UNIVERSITY  OF  CALIFORN:; 
AT    LOS  ANGELES 


AN 

IIVTRODUCTION 

TO 

ALGEBRA, 

WITH 

NOTES  AND  OBSERVATIONS ; 

DESIGNED  FOR  THE 

JSE  OF  SCHOOLS  AND  PLACES  OF  PUBLIC  EDUCATIO> 

TO  WHICH  13  ADDED 

AN  APPENDIX, 

ON  THE 

APPLICATION  OF  ALGEBRA  TO  GEOMEl'Rl 


BY  JOHN  BONNYCASTI.E, 

Professor  of  Mathematics  in  the  Royal  Military  Acadenir,  Woolwich. 

FOURTH  NEW- YORK,  FROM  THE  LAST  LONDON  EDITION. 


EVISED,  CORK£CTED,  AND  ENLARGED,  WITH  A  VARIETY  OF  EXAMPJLES    AKli 
MANY  OTHER  USEFUL  ADDITIONS,  * 

BY  JAMES  RYAN, 

Author  of  "  An  Elementary  Treatise  on  Algebra,  Theoretical  and 
Practical,"  &c. 


-Ingenuas  didicisse  fideliter  artes 

Emollit  mores,  nee  sinit  esse  feros.        Ovid. 


NEW-YORK : 

PUBLISHED  BY  EVERT  DUyCKINCK,  AND  COLLINS  &  HANNAH. 


^637        @«2'- 


Southern  District  of  New- York,  ss. 

BE  IT  REMEMBERED,  That  on  the  28th  day  of  December,  in  the  forty  ^ 
sixth  year  of  the  Independence  of  the  United  States  of  America,  George 
Long,  of  the  said  District,  hath  deposited  \n  this  office  the  title  of  a  Book, 
the  right  whereof  he  claims  as  Proprietor,  in  the  words  following,  to  wit : 

•'  An  Introduction  to  Algebra,  with  Notes  and  Observations ;  designed/or 
the  Use  of  Schools  and  places  of  Public  Education.  To  which  is  added  an 
Appendix,  on  the  Application  of  Algebra  to  Geometry.  By  John  Bonny- 
castle,  Professor  of  Mathematics  in  the  Royal  Military  Academy,  Woolwich. 
Fourth  New-York,  from  the  Last  London  Edition.  Revised,  corrected,  and 
enlarged,  with  a  variety  of  Examples,  and  many  other  useful  Additions,  by 
James  Ryan,  Author  of  an  Elementary  Treatise  on  Algebra,  Tlieoretical  and 
Practical,  &c. 

Ingenuas  didicisse  fideliter  artes 

Emollit  mores,  nee  sinit  esse  feros.  Ot'id." 

In  conformity  to  the  Act  of  Congress  of  the  United  States,  entitled  "  An 
Act  or  the  encouragement  of  Learning,  by  securing  the  copies  of  Maps, 
Charts,  and  Books,  to  the  Authors  and  Proprietors  of  such  copies,  during  the 
time  therein  mentioned,"  and  also  to  an  Act,  entitled  "  An  Act,  supplemen- 
tary to  an  Act,  entitled,  An  Act  for  the  encouragement  of  Learning,  by  se- 
curing the  copies  of  Maps,  Charts,  and  Books,  to  the  Authors  and  Proprie- 
tors of  such  copies  during  the  times  therein  mentioned,  and  extending  the 
benefits  thereof  to  the  arts  of  Designing,  Engraving,  and  Etching  Historical 
and  other  Prints." 

JAMES  DILL, 
aerk  of  the  Southern  District  ofjyew-Yorh. 


W.  E.  DEAN,  PRINTER. 


Sciences  ^     v 
Library    Q>s/A 


PREFACE 


The  powers  of  the  mind,  like  those  of  the  body,  are  m- 
creased  by  frequent  exertion  ;  application  and  industry 
supply  the  place  of  genius  and  invention  ;  and  even  the 
creative  faculty  itself  may  be  strengthened  and  improved 
by  use  and  perseverance.  Uncultivated  nature  is  uniform" 
ly  rude  and  imbecile,  it  being  by  imitation  alone,  that  we 
at  first  acquire  knowledge,  and  the  means  of  extending  its 
bounds.  A  just  and  perfect  acquaintance  with  the  simple 
elements  of  science,  is  a  necessary  step  towards  our  future 
progress  and  advancement  ;  and  this,  assisted  by  laborious 
investigation  and  habitual  inquiry,  will  constantly  lead  to 
eminence  and  perfection. 

Books  of  rudiments,  therefore,  concisely  written,  well 
digested,  and  methodically  arranged,  are  treasures  of  ines- 
timable value  ;  and  too  many  attempts  cannot  be  made  to 
render  them  perfect  and  complete.  When  the  first  prin- 
ciples of  any  art  or  science  are  firmly  fixed  and  rooted  in 
the  mind,  their  application  soon  becomes  easy,  pleasant, 
and  obvious  ;  the  understanding  is  delighted  and  enlarged ; 
we  conceive  clearly,  reason  distinctly,  and  form  just  and 
satisfactory  conclusions.  But,  on  the  contrary,  when  the 
mind,  instead  of  reposing  on  the  stabiUty  of  truth  and  re- 
ceived principles,  is  wandering  in  doubt  and  uncertainty, 
our  ideas  will  necessarily  be  confused  and  obscure  ;  and 
every  step  we  take  must  be  attended  with  fresh  difficulties 
and  endless  perplexity. 


im^GB 


iv  PREFACE. 

That  the  grounds,  or  fundamental  parts,  of  every  sci 
ence,  are  dull  and  unentertaining,  is  a  complaint  univer- 
sally made,  and  a  truth  not  to  be  denied  ;  but  then,  what 
is  obtained  with  difficulty  is  usually  remembered  with  ease  i 
and  what  is  purchased  with  pain  is  often  possessed  with 
pleasure.  The  seeds  of  knowledge  are  sown  in  every  soil, 
but  it  is  by  proper  culture  alone  that  they  are  cherished 
and  brought  to  maturity.  A  few  years  of  early  and  assi- 
duous application  never  fails  to  procure  us  the  reward  of 
our  industry :  and  who  is  there,  who  knows  the  pleasures 
and  advantages  which  the  sciences  afford,  that  would  think 
his  time,  in  this  case,  misspent,  or  his  labours  useless  : 
riches  and  honours  are  the  gifts  of  fortune,  casually  be- 
stowed, or  hereditarily  received,  and  are  frequently  abus. 
ed  by  their  possessors  ;  but  the  superiority  of  wisdom  and 
knowledge  is  a  pre-eminence  of  merit,  which  originates 
with  the  man,  and  is  the  noblest  of  all  distinctions. 

Nature,  bountiful  and  wise  in  all  things,  has  provided  us 
with  an  infinite  variety  of  scenes,  both  for  our  instruction 
and  entertainment  ;  and,  like  a  kind  and  indulgent  parent,, 
admits  all  her  children  to  an  equal  participation  of  her  bless- 
ings. But,  as  the  modes,  situations,  and  circumstances 
of  life  are  various,  so  accident,  habit,  and  education,  have 
each  their  predominating  influence,  and  give  to  every  mind 
its  particular  bias.  Where  examples  of  excellence  are 
wanting,  the  attempts  to  attain  it  are  but  few ;  but  emi- 
nence excites  attention  and  produces  imitation.  To  raise 
the  curiosity,  and  to  awaken  the  listless  and  dormant  pow- 
ers of  younger  minds,  we  have  only  to  point  out  to  them  a 
valuable  acquisition,  and  the  means  of  obtaining  it ;  the 
active  principles  are  immediately  put  into  motion,  and  the 
certainty  of  the  conquest  is  ensured  from  a  determination 
to  conquer. 

But,  of  all  the  sciences  which  serve  to  call  forth  this 
spirit  of  enterprize  and  inquiry,  there  are  none  more  emi- 
nently useful  than  Mathematics.  By  an  early  attachment 
to  these  elegant  and  sublime  studies,  we  acquire  a  habit  of 
reasoning,  and  an  elevation  of  thought,  which  fixes  the 
mind,  and  prepares  it  for  every  other  pursuit.     From  i^ 


PREFACE.  V 

:'e\v  simple  axioms,  and  evident  principles,  we  proceed 
gradually  to  the  most  general  propositions,  and  remote  an«> 
alogies  ;  deducing  one  truth  from  another,  in  a  chain  of 
argument  well  connected  and  logically  pursued  ;  which 
brings  us  at  last,  in  the  most  satisfactory  manner,  to  the 
conclusion,  and  serves  as  a  general  direction  in  all  oar  in- 
quiries after  truth. 

And  it  is  not  only  in  this  respect  that  mathematical  learn- 
ing is  so  highly  valuable  ;  it  is  likewise  equally  estimable 
for  its  practical  utility.  Almost  all  the  works  of  art  and 
devices  of  man  have  a  dependence  upon  its  principles,  and 
are  indebted  to  it  for  their  origin  and  perfection.  The 
cultivation  of  these  admirable  sciences  is,  therefore,  a 
thing  of  the  utmost  importance  and  ought  to  be  considered 
as  a  principal  part  of  every  liberal  and  well-regulated  plan 
of  education.  They  are  the  guide  of  our  youth,  the  per- 
fection of  our  reason,  and  the  foundation  of  every  great 
and  noble  undertaking. 

From  these  considerations,  I  have  been  induced  to  com- 
pose an  introductory  course  of  mathematical  science  ;  and 
from  the  kind  encouragement  which  I  have  hitherto  re- 
ceived, am  not  without  hopes  of  a  continuance  of  the  same 
candour  and  approbation.  Considerable  practice  as  a 
teacher,  and  a  long  attention  to  the  difficulties  and  obstruc- 
tions which  retard  the  progress  of  learners  in  general, 
have  enabled  me  to  accommodate  myself  the  more  easily  to 
their  capacities  and  understandings.  And  as  an  earnest  de- 
sire of  promoting  and  diffusing  useful  knowledge  is  the 
chief  motive  for  this  undertaking,  so  no  pains  or  attention 
shall  be  wanting  to  make  it  as  complete  and  perfect  as  pos- 
sible. 

The  subject  of  the  present  performance  is  Algebra  : 
which  is  one  of  the  most  important  and  useful  branches  of 
those  sciences,  and  m.ay  be  justly  considered  as  the  key  to 
all  the  rest.  Geometry  delights  us  by  the  simplicity  of  ity 
principles  and  the  elegance  of  its  demonstrations  ;  Arith- 
metic is  confined  to  its  object,  and  partial  in  its  application  ; 
but  Algebra  or  the  Analytic  Airl,  is  general  and  comprehen- 
sive, and  may  be  applied  with  success  in  all  cases  wher*: 
a2 


n  PREFACE. 

truth  is  to  be  obtained  and  proper  data  can  be  esta- 
blished. 

To  trace  this  science  to  its  birth,  and  to  point  cut  the 
\  arious  aherations  and  improvements  it  has  undergone  in 
its  progress,  would  far  exceed  the  limits  of  a  preface.*  It 
will  be  sufficient  to  observe,  that  the  invention  is  of  the 
highest  antiquity,  and  has  challenged  the  praise  and  admi- 
ration of  all  ages.  Diophantus,  a  Greek  mathematician,  of 
Alexandria  in  Egypt,  who  flourished  in  or  about  the  third 
century  after  Christ,  appears  to  have  been  the  first,  among 
the  ancients,  who  applied  it  to  the  solution  of  indetermi- 
nate or  unlimited  problems  ;  but  it  is  to  the  moderns  that 
we  are  principally  indebted  for  the  most  curious  refine- 
ments of  the  art,  and  its  great  and  extensive  usefulness  in 
every  abstruse  and  difficult  inquiry.  Kem^ton,  Maclaurin^ 
Sannderson,  Simpson,  and  Emerson^  among  our  own  country- 
men, and  Clairaut,  Eider,  Lagrange,  and  Lacroix,  on  the 
continent,  are  those  who  have  particularly  excelled  in  this 
respect  ;  and  it  is  to  their  works  that  I  would  refer  the 
young  student,  as  the  patterns  of  elegance  and  perfection. 

The  following  compendium  is  formed  entirely  upon  the 
model  of  those  writers,  and  is  intended  as  a  useful  and  ne- 
cessary introduction  to  them.  Almost  every  subject,  which 
belongs  to  pure  Algebra,  is  concisely  and  distinctly  treated 
of;  and  no  pains  have  been  spared  to  make  the  whole  as 
easy  and  intelligible  as  possible.  A  great  number  of  ele- 
mentary books  have  already  been  written  upon  this  sub- 
ject ;  but  there  are  none,  which  I  have  yet  seen,  but  what 
appear  to  me  to  be  extremely  defective.  Besides  being 
totally  unfit  for  the  purpose  of  teaching,  they  are  generally 
calculated  to  vitiate  the  taste,  and  mislead  the  judgment- 
A  tedious  and  inelegant  method  prevails  through  the  whole, 
so  that  the  beauty  of  the  science  is  generally  destroyed  by 
the  clumsy  and  awkward  manner  in  which  it  is  treated  : 
and  the  learner,  when  he  is  afterwards  introduced  to  some 

*  Those  who  are  desirous  of  a  knowledge  of  this  kind,  may  consult  the  In 
troduction  to  my  Treatise  on  Algebra  ;  where  they  will  6nd  a  regular  his 
torical  detail  of  the  rise  and  progress  of  the  science,  from  its  first  rude  beg'n 
piBgs  to  the  present  times.  * 


PREFACE.  tB 

of  our  best  writers,  is  obliged,  in  a  great  measure,  to  un- 
learn and  forget  every  thing  which  he  has  been  at  so  much 
pains  in  acquiring. 

There  is  a  certain  taste  and  elegance  in  the  sciences,  as 
well  as  in  every  branch  of  polite  literature,  which  is  only 
to  be  obtained  from  the  best  authors,  and  a  judicious  use 
of  their  instructions.  To  direct  the  student  in  his  choice 
of  books,  and  to  prepare  him  properly  for  the  advantages 
he  may  receive  from  them,  is  therefore  the  business  of 
every  writer  who  engages  in  the  humble,  but  useful  task 
of  a  preliminary  tutor.  This  information  I  have  been 
careful  to  give,  in  every  part  of  the  present  performance, 
where  it  appeared  to  be  in  the  least  necessary  ;  and, 
though  the  nature  and  confined  limits  of  my  plan  admitted 
not  of  diffuse  observations,  or  a  formal  enumeration  of  par- 
ticulars, it  is  presumed  nothing  of  real  use  and  importance 
has  been  omitted.  My  principal  object  was  to  consult  the 
ease,  satisfaction,  and  accommodation  of  the  learner;  and 
the  favourable  reception  the  work  has  met  with  from  the 
public,  has  afforded  me  the  ^-ratification  of  believing  that 
rny  labours  have  not  been  unsuccessfully  employed. 


ADVERTISE.^IENT, 


The  present  performance  having  passed 
through  a  number  of  editions  since  the  time 
of  its  first  pubhcation,  without  any  material 
alterations  having  been  made,  either  with  re- 
spect to  its  original  plan,  or  the  manner  in 
which  it  was  executed,  I  have  been  induced, 
from  the  flattering  approbation  it  has  constant- 
ly received,  to  undertake  an  entire  revision 
of  the  work ;  and,  by  availing  myself  of  the 
improvements  that  have  been  subsequently 
made  in  the  science,  to  render  it  still  more  de- 
serving the  public  favour. 

In  its  present  state,  it  may  be  considered  as 
a  copious  abridgment  of  the  most  practical 
and  useful  parts  of  my  larger  work,  entitled. 
A  Treatise  on  Algebra,  in  2  vols.  8vo.  publish- 
ed in  18i3  ;  from   which,   except  in  certain 
cases,  where  a  different  mode  of  proceeding 
appeared  to  be  necessary,  it  has  been  chiefly 
compiled :  great  care  having  been  taken,  at 
ihe  same  time,  to  adapt  it,  as  much  as  possible 
to  the  wants  of  the  learners,  and  the  genera 
purposes  of  instruction,  agreeably  to  the  de 
-isfn  with  which  it  was  first  written. 


IX 


With  this  view,  as  well  as  in  compliance 
with  the  wishes  of  several  inteUigent  teachers, 
I  have  also  been  led  to  subjoin  to  it,  by  way  of 
an  Appendix,  a  small  tract  on  the  application 
of  Algebra  to  the  solution  of  Geometrical 
Problems  ;  which,  it  is  hoped,  will  prove  ac- 
ceptable to  such  classes  of  students  as  ma} 
not  have  an  opportunity  of  consulting  more 
voluminous  and  expensive  works  on  this  inte- 
resting branch  of  the  science. 

JOHN  BONNYCASTLE. 


Royal  Military  Academy, 
Woolwich. 
October  22,  1815, 


ADVERTISEMEXTS 

TO 

THE  SECOJVD  NEW-YORK  EDITION 


It  would  be  superfluous  to  advance  any  thing 
ni  commendation  of ''  Bonnycastle's  Introduc- 
tion to  Algebra,"  as  the  number  of  European 
editions,  and  the  increase  of  demand  for  it 
since  its  publication  in  this  country,  are  suffi- 
cient proofs  of  its  great  utility. 

But  to  make  it  universally  useful  both  to 
the  tutor  and  scholar,  I  have  given  in  this  edi- 
tion, the  answers  that  were  omitted  by  the  Au- 
thor in  the  original. 

In  the  course  of  the  work,  particularly  in 
Addition,  Subtraction,  Multiplication,  Divi- 
sion, Fractions,  Simple  Equations,  and  Quad- 
ratics, I  have  added  a  great  variety  of  prac- 
tical examples,  as  being  essentially  necessary 
to  exercise  young  students  in  the  elementary 
principles. 

Several  new  rules  are  introduced,  those  of 
principal  note  are  the  following :  Case  12. 
Surds,  containing  two  rules  for  fiinding  any 
root  of  a  Binomial  Surd,  the  Solution  of  Cu- 
bics  by  Converging  Series,  the  Solution  of  Bi- 
quadratics by  Simpson's  and  Euler's  methods  • 


XI 

ail  these  rules  ai*e  investigated  in  the  plainesl 
manner  possible,  with  notes  and  remarks,  in- 
terspersed throughout  the  work,  containing 
some  very  useful  matter. 

There  is  also  given  all  the  Diophantine  Ana- 
lysis, contained  in  Bonnycastle's  Algebra,  Vol, 
1.  8vo.  1820,  being  a  methodical  abstractor 
this  part  of  the  science,  which  comprehends 
most  of  the  methods  hitherto  known  for  re- 
solving problems  of  this  kind,  and  will  bo 
found  a  ready  compendium  for  such  readers 
as  may  acquire  some  knovvledge  of  the  Ana- 
lytic Art. 

JAMES  RYAN, 

New-York,  Jan,  1.  1822. 


COXTENTS. 


DEFINITIONS 

Addition 

Subtraction    . 

MiiltiplicatiOQ 

Division 

Algebraic  Fractions 

Involution,  or  tiie  Raising  of  Powers 

Evolution,  or  the  Extraction  of  Roots 

Of  Irrational  Quantities,  or  Surds     . 

Of  Arithmetical  Proportion  and  Progression 

Of  Geometrical  Proportion  and  Progression 

Of  Equations  .... 

Of  the  Resolution  of  Simple  Equations 

Miscellaneous  Questions 

Of  Quadratic  Equations 

Questions  producing  Quadratic  Equations 

Of  Cubic  Equations 

Of  the  Solution  of  Cubic  Equations 

Of  the  Solution  of  Cubic  Equations  by  Converging  Series 

Of  the  Resolution  of  Biquadratic  Equations 

To  find  the  Roots  of  Equations  by  Approximation 

To  find  the  Roots  of  Exponential  Equations 

Of  the  Binomial  Theorem     . 

Of  the  Indeterminate  Analysis 

Of  the  Diophanline  Analysis 

Of  the  Summation  and  Interpolation  of  Series 

Of  Logarithms  .... 

Multiplication  by  Logarithms 

'Division  by  Logarithms 

The  Rule  of  Three  by  Logarithms 

Involution,  by  Logarithms     . 

Evolution,  by  Logarithms      . 

A  Collection  of  Miscellaneous  Questions 

Appendix,  on  the  Application  of  Algebra  to  Geometry 


Pagj 

8 

12 

U 

19 

27 

U 

47 

5* 

87 

91 

99 

101 

113 

128 

139 

145 

150 

155 

163 

176 

182 

184 

19Q 

201 

234 

257 

269 

272 

^74 

2/6 

278 

281 

28- 


algebra; 


ALGEBRA  is  the  science  which  treats  of  a  general  me= 
thod  of  performing  calculations,  and  resolving  mathemati" 
cal  problems,  by  means  of  the  letters  of  the  alphabet. 

Its  leading  rules  are  the  same  as  those  of  arithmetic  i 
and  the  operations  to  be  performed  are  denoted  by  the  fol- 
lowing characters  : 

-f-  plus  or  more,  the  sign  of  addition ;  signifying  that  thc 
quantities  between  which  it  is  placed  are  to  be  added  toge- 
ther. 

Thus,  a-f  6  shows  that  thc  number,  or  quantity,  repre- 
sented by  b,  is  to  be  added  to  that  represented  by  a  ;  and 
is  read  a  plus  6. 

-  minus,  or  less,  the  sign  of  subtraction  ;  signifying  that 
the  latter  of  the  two  quantities  between  which  it  is  placed 
is  to  be  taken  from  the  former. 

Thus  a — 6  shows  that  the  quantity  represented  by  h  is 
to  be  taken  from  that  represented  by  a  :  and  is  read  a  mi* 
nus  b. 

Also,  a^b  represents  the  difference  of  the  two  quanti- 
ties a  and  6,  when  it  is  not  known  which  of  them  is  the 
greater. 

X  iniOf  the  sign  of  multiplication  ;  signifying  that  tho 
quantities  between  which  it  is  placed  are  to  be  multiplied 
together. 

Thus,  aXb  shows  that  the  quantity  represented  by  a  is 
to  be  multiplied  by  that  represented  by  6  ;  and  is  read  a 
into  6. 

The  multiplication  of  simple  quantities  is  also  frequently 
denoted  by  a  point,  or  by  joining  the  letters  together  in  the 
form  of  a  word. 


2  DEFINITIONS, 

Thus,  aXb^a  .  £,  and  afe,  all  signify  the  product  of  ti 
and  b ;  also,  3  Xa,  or  3a,  is  the  product  of  3  and  a  :  and  ie 
read  3  times  a. 

-r-  bjjf  the  sign  of  division  ;  signifying  that  the  former  of 
the  two  quantities  between  which  it  is  placed  is  to  be  divid- 
ed by  the  latter. 

Thus,  a-r-6,  shows  that  the  quantity  represented  by  a  is 
to  be  divided  by  that  represented  by  b  ;  and  is  read  a  by  b, 
or  a  divided  by  6. 

Division  is  also  frequently  denoted  by  placing  one  of 
the  two  quantities  over  the  other,  in  the  form  of  a  frac- 
tion. 

Thus,  6-r-a  and  -  both   signify  the    quotient   of  b  di- 
ft 

vided  by  a  ;  and  — -r—  signifies  that  a  -  6  is  to  be  divided 

by  a-^-c. 

=  equal  to,  the  sign  of  equality  ;  signifying  that  the 
quantities  between  which  it  is  placed  are  equal  to  each 
other. 

Thus,  x=a-\-b  shows  that  the  quantity  denoted  by  x  is 
equal  to  the  sum  of  the  quantities  a  and  6  ;  and  is  read  x 
equal  to  a  plus  b. 

Any  two  algebraic  expressions  are  said  to  be  identical^ 
when  they  are  of  the  same  value,  for  all  the  values  of  the 
letters  of  which  they  are  composed. 

*  Thus,  (x-|-a)  X  (x-a)  =x^-a2,  whatever  numeral 
values  may  be  given  to  the  quantities  represented  by  a 
and  a. 


»  Woodhouse,  in  his  principles  of  Analytical  calculation,  says  that  x^-^a,^ 
is  not  generally  c=  (x — a),  (a-f-x) :  for  instance,  the  particular  case  of  a:  <=  a 
is  to  be  excluded;  the  proof  essentially  demanding  this  circumstance  to  wit, 
that  X — a  be  a  quantity,  or  that  x  be  greater  than  a.  Euler  calls  x — 1  =  x~\ 
an  identical  equation  ;  and  shows  that  x  is  indeterminate,  or  that  any  number 
whatever  may  be  substituled  for  it ;  §ee  Euler's  Algebra,  page  289,  Vol,  I. 

F-- 


DEFINITIONS.  ^ 

>  greater  than,  the  sign  of  majority  ;  signifying  that  the 
former  of  the  two  quantities  between  which  it  is  placed  is 
greater  than  the  latter. 

Thus  a>b  shows  that  the  quantity  represented  by  a  is 
greater  than  that  represented  by  b  ;  and  is  read  a  greater 
than  b. 

<  less  thauy  the  sign  of  minority  ;  signifying  that  the  for* 
mer  of  the  two  quantities  between  which  it  is  placed  is  less 
than  the  latter. 

Thus,  a<6  shows  that  the  quantity  represented  by  a 
is  less  than  that  represented  by  6  ;  and  is  read  a  less  than 
b. 

:  aSf  or  to,  and  :  :  so  is,  the  signs  of  an  equality  of  ra- 
tios ;  signifying  that  the  quantities  between  which  they  are 
placed  are  proportional. 

Thus,  a:  b  :  :  c  :  d  denotes  that  a  has  the  same  ratio 
to  h  that  c  has  to  c/,  or  that  a,  b,  c,  rf,  are  proportionals  ; 
and  is  read,  us  a  is  to  6  so  is  v  to  t/,  or  a  is  to  6  as  c  is 
tod, 

^  the  radical  siifw,  signifying  that  the  quantity  be- 
fore which  it  is  placed  is  to  have  some  root  of  it  extract- 
ed. 

Thus,  y/a  is  the  square  root  of  «  ;  l/a  is  the  cube  root 
of  a  ;  and  ^a  is  the  fourth  root  of  a  ;  &c. 

The  roots  of  quantities  are  also  represented  by  figures 
placed  at  the  right  hand  corner  of  them,  in  the  form  of  a 
fraction. 

Thus,  a2  is  the  square  root  of  a  ;  d^  is  the   cube   root 
± 
of  a  ;    and  a''  is  the  nth  root  of  a,  or  a  root  denoted  by  any 
number  n. 

In  like  manner,  a^  is  the  square  of  a  ;  a^  is  the  cube  of 
•X  ;  and  a""'  is  the  mth  power  of  a,  or  any  power  denoted  by 
the  number  m. 

CO   is  the  sign  of  infinity,  signifying  that  the  quantity 


4  DEFINITIONS. 

standing  before  it  is  of,  an  unlimited  value^  or  greater  tbsK 
any  quantity  that  can  be  assigned. 

The  coefficient  of  a  quantity  is  the  number  or  letter 
which  is  prefixed  to  it. 

Thus,  in  the  quantities  Sb,  —  |6,  3  and  —  f  are  the  co- 
efficients of  b  ;  and  a  is  the  coefficient  of  x  in  the  quantity 
ax» 

A  quantity  without  any  coefficient  prefixed  to  it  is  sup- 
posed to  have  1  or  unity  ;  and  when  a  quantity  has  no  sign 
before  it,  -|-  is  always  understood. 

Thus,  a  is  the  same  as  +  a,  or  +  1«  ;  and  —a  is  the 
same  as  —  la. 

A  term  is  any  part  or  member  of  a  compound  quan- 
tity, which  is  separated  from  the  rest  by  the  signs  -f* 
or  —. 

Thus,  a  and  b  are  the  terms  of  a  +  6  ;  and  3a, —  26. 
and  +  5cdj  are  the  lerm.s  of  3(i-  26+6cc/. 

In  like  manner,  the  terms  of  a  product,  fraction,  or  pro- 
portion, are  the  several  parts  or  quantities  of  which  they 
are  composed. 

Thus,  a  and  b  are  the  terms  of  a6,  or  of^  ;  and  c,  b,  c. 
df  are  the  terms  of  the  proportion  a:  b  :  '.  r.  :  d. 

A  factor  is  one  of  the  terms,  or  multipliers  which  form 
the  product  of  two  or  more  quantities. 

Thus,  a  and  b  are  the  factors  of  ab  ;  also,  2,  a,  and  t^, 
are  the  factors  of  2aP  ;  and  a  — a;  and  b—x  are  the  factors 
of  the  product  (a  — rr)  X  (6— x). 

A  composite  number,  or  quantity,  is  that  which  is  pro- 
duced by  the  multiplication  of  two  or  more  terms  or  fac- 
tors. 

Thus,  6  is  a  composite  nunjber,  formed  of  the  factors  2 
and  3,  or  2x3  ;  and  3abc  is  a  composite  quantity,  the  fac- 
tors of  which  are  3,  a,  b,  c. 

Like  quantities,  are  those  which  consist  of  the  same  let- 
ters or  combinations  of  letters  ;  as  a  and  3a,  or  bab  and 
7a6,  or  2a^b  and  9a'b. 


DEFINITIONS.  5 

Unlike  quantities,  are  those  which  consist  of  different  let- 
ters, or  combinations  of  letters  ;  as  a  and  6,  or  3a  and  a% 
or  5ab^  and  7a-6. 

Given  quantities,  are  such  as  have  known  values,  and  are 
generally  represented  by  some  of  the  first  letters  of  the  al- 
phabet ;  as  a,  b,  c,  rf,  &c. 

Unknown  quantities,  are  such  as  have  no  fixed  values, 
and  are  usually  represented  by  some  of  the  final  letters  of 
the  alphabet ;  as  x,  y,  z» 

Simple  quantities,  are  those  which  consist  of  one  term 
only;  as  3a,  bah,  — 8a^6,  &c. 

Compound  quantities,  are  those  which  consist  of  several 
terms  ;  as  2a-\-b,  or  3a  — 2c,  or  a+26  — 3c,  &c. 

Positive,  or  affirmative  quantities,  are  those  which  are  to 
be  added  ;  as  a,  or  +a,  or  +3a6,  &c. 

Negative  quantities  are  those  which  are  to  be  subtract- 
ed ;  as  — a,  or  —  3a6,  or  —  7a6^,  &c. 

Like  signs,  are  such  as  are  all  positive,  or  all  negative ; 
as  -f-  and  +,  or  —  and  —. 

Unlike  signs,  are  when  some  are  positive  and  others  ne- 
gative ;  as  +  and  — ,  or  —  and  -1-. 

A  monomial,  is  a  quantity  consisting  of  one  term  only  :  as 
a,  26,  —  3a^6,  &c. 

A  binomial,  is  a  quantity  consisting  of  two  terms,  as  a+&, 
or  a  —  6  ;  the  latter  of  which  is,  also,  sometimes  called  a 
residual  quantity. 

A  trinomial,  is  a  quantity  consisting  of  three  terms,  as 
a+26 — 3c;  a  quadrinomial  of  four,  as  a— 26+3c— dt 
and  a  polynomial,  or  multinomial,  is  that  which  has  many 
terms. 

The  power  of  a  quantity,  is  its  square,  cube,  biquadrate, 
&c.  ;  called  also  its  second,  third,  fourth  power,  &c.  ;  as 
a^,  a^,  a*,  &c. 

The  index,  or  exponent  of  a  quantity,  is  the  numb^V 
%'hich  denotes  its  power  or  root. 


b2 


a  DEFINITIONS. 

Thus,  —  1  is  the  index  of  a     »,  2  is  the  index  of  a% 

and  i  of  a2  or  ^a. 

When  a  quantity  appears  without  any  index,  or  exponent, 
it  is  always  understood  to  have  unity,  or  1. 

Thus,  a  is  the  same  as  a\  and  ^cc  is  the  same  as  2z^  : 
the  1,  in  such  cases,  heing  usually  omitted. 

A  rational  quantity,  is  that  which  can  be  expressed  in 
finite  terms,  or  without  any  radical  sign,  or  fractional  in- 
dex; as  a,  or  fa,  or  5u,  &c. 

*  An  irrational  Quantity,  or  Surd,  is  that  of  which  the 
value  cannot  be  accurately  expressed  in  numbers,  as  the 
square  roots  of  2,  .S,  5.  Surds  are  commonly  expressed 
by  means  of  the  radical  sign  ^  ;  as  \/2,  ^a,  {/a^i  or  a 


JL        2 


fractional  index;  as  2'-,a3,  &c. 

A  square  or  cube  number,  &c.  is  that  which  has  an  exact 
square  or  cube  root,  &c. 

Thus,  4  and  y%a-  are  square  numbers  ;  and  64  and 
-^a^  are  cube  numbers,  &c. 

A  measure  of  any  quantity,  is  that  by  which  it  can  be  di- 
vided without  leaving  a  remainder. 

Thus,  3  is  a  measure  of  6,  7a  is  a  measure  of  35a,  and 
dah  of  27  a%'. 

Commensurable  quantities,  are  such  as  can  be  each 
divided  by  the  same  quantity,  without  leaving  a  remain- 
der. 

Thus,  6  and  8,  2  v/2  and  3^2,  5o-6  and  7o6^  are  com- 
mensurable quantities ;  the  common  divisors  being  2,  ^2,. 
and  ab. 

Incommensurable  quantities,  are  such  as  have  no  com- 
mon measure,  or  divisor,  except  unity. 

Thus,  15  and  16,  v^  2  and  ^3,  and  a-^-b  and  a^-^b'  are 
incommensurable  quantities. 


*  This  definition  of  a  Surd,  or  irrational  Quantity,  is  due  to  Robert  Adrain. 
LL.D.,  Professor  of  Mathematics  and  Natural  Philosophy  in  Columbia  Col- 
lege New-York ;  vfho  had  first  published  it  in  his  edition  of  Hutlon's  course 
of  Mathematics.  Er 


DEFINITIONS.  ^ 

A  multiple  of  any  quantity,  is  that  which  is  some  exact 
2iuniber  of  times  that  quantity. 

Thus,  12  is  a  multiple  of  4,  liia  is  a  multiple  of  3a,  and 
20a-62  of  5ab. 

The  reciprocal  of  any  quantity,  is  that  quantity  inverted, 
or  unity  divided  by  it. 

Thus,  the  reciprocal  of  of,  or  f ,  is  i  ;  and  the  reciprocal 

of.-  i=  I- 

A  function  of  one  or  more  quantities,  is  an  expression 

into  which  those  quantities  enter,  in  any  manner  whatever, 

cither  combined,  or  not,  with  known  quantities. 

Thus,  a -.2a:,  ax-{-3x\  2x-a  {a^^x^y ,  fla;"',  a%  &c.j 

are  functions  of  x  ;  and  axy-^-bx^,  ay-^x  {ax^ •^hy^)'- ^ 
&c.  are  functions  of  a:  and  y. 

A  vinculum,   is   a  bar ,  or   parenthesis  (  ),  made 

use  of  to  collect  several  quantities  into  one. 

Thusa-|-6Xc,  or  (a-{-6)  c,  denotes  that  the  compound 
quantity  a-\rh  is  to  be  multiphed  by  the  simple  quantity  c  ; 
and  ^ah-^-c"",  or  (afe  +  c^)?,  is  ihe  square  root  of  the  com- 
nound  quantity  ah-\-c^, 

Vractical  Examplesfor  computing  the  numeral  V^ahies  ofva. 
rioKS  Algebraic  Expressions,  or  Combinations  of  Letters, 

Supposing  a~Qj  b=5,  c=4,  tZ~l,  and  €=0. 
Then 

1.  a24-2a6-c  +  f^=36+60-4+l=93, 

2.  2a='--3ft^6+c^=432-540-f 64~~44. 

3.  a2x«4-&  —  2a6c==36>a  1-240=160. 

4.  2a yl^Vc+^ggc-f  €"=12X14-8=^20. 

V.  3a^2^c\ov_3a  (2ac+cy=18y64=]44., 

6.  ^¥a^-v/2aH^2~^72--v/64=^72^8=V64^~ 

2a-f3c_^ 46c      __I2+12      80 _^  ,  SO 

'  6d-|-4e      V2ac+7a"       6  +  0        v'48+^6'~*  6       S 

5=14. 


t?  ADDITION. 

Required  the  numeral  values  of  the  following  quantities  ; 
supposing  a,  6,  c,  d,  c,  to  be  6,  6,  4,  1,  and  0,  respectively, 
as  above. 

1.  2a'-f36c-5£/=127 

2.  50^6- lOa62-}-2e=-f.OO 

3.  7a2-{-6-.cX<i+e=253 

4.  5^a6-{-^/2_2a6-e2=  -7.613S75 
a   .  ,     a  — 6 


5.  _Xd —  4.2«2e= 

c  o 


6.  3v/£+?av/2a  +  6-i/=l4 

7.  a^a2+62+3&cy^_62=245.8589S62 

S.  3a26+yc2+^2^^:p^=542.884499I 
*  3a-c  2a+c  '** 

ADDITION. 

Addition  is  the  connecting  of  quantities  tegether  by 
means  of  their  proper  signs,  and  incorporating  such  as  are 
like,  or  that  can  be  united,  into  one  sum  ;  the  rule  for  per- 
forming which  is  commonly  divided  into  the  three  following 

cases.* 

CASE  I. 

When  the  (Quantities  are  like,  and  have  like  signs, 

RULE. 

Add  all  the  coefficients  of  the  several  quantities  together, 
tmd  to  their  sum  annex  the  letter  or  letters  belonging  to 
each  term,  prefixing,  when  necessary,  the  common  sign. 

*  The  term  Addition,  which  is  generally  used  to  denote  this  rule,  is  too 
scanty  to  express  the  nature  of  the  operations  that  are  to  be  performed  in  it  : 
which  are  sometimes  those  of  addition,  and  sometimes  subtraction,  according' 
as  the  quantities  are  negative  or  positive.  It  should,  therefore,  be  called  by 
^ne  nanae  signifying  incorporation,  or  striking  a  balance  .;  in  which  case, 
the  incongruity  here  mentioned  vtould  be  removed. 


ADDITION. 

Examples. 

3a 
5a 

\a 

i2a 

—Zax 

—btxx 

—  nx 

—  2ax 

—  ''tax 

2b-^'Sy 
bb-\'ly 
b+2y 
8h+y 
46+4t/ 

28a 

—  \^ax 

206+172; 

2ay 
bay 
4ay 
lay 
Ibay 

-2b  f 

-bf 

—  &by^ 

-  bf 

-^ISby^ 

a-2x2 
a-6x- 

4a— x2 
3a -.5x" 
7a -x^ 

Siay 

16a- 15x' 

Sax^ 
2ax^ 

12ax2 
9ax^ 

lOax- 

lx—4y 
x-^Sy 

3r— V 

x-Sy 
4x — V 

iGx-lly 

2a+x2 
3a4-x2 
a  +  2x" 
9a+3x- 
4a+x2 

max^ 

I9a+8x2 

CASE  II. 
When  the  (Quantities  are  like,  but  have  unlike  signs. 

RULE. 

Add  all  the  affirmative  coefficients  into  one  sum,  and 
those  that  are  negative  into  another,  when  there  are  se- 


10  ADDITION. 

veral  of  the  same  kind  :  then  subtract  the  least  of  these 
sums  from  the  greatest,  and  to  the  difference  prefix  the 
sign  of  the  greater,  annexing  the  common  letter  or  letters 
3.S  before. 


EXAMPLES. 


— 3a  2a-^3x  6x-{-Bay 

+7a  —7^4-5x2  —Sx+2ay 

X — Gay 
+a— 3x2  2x-{-ay 


+  lla  -^la     *  6x+2ay 


—2a' 

3a2/-7 

— 3fl6-{-7x 

— 3a2 

-«i/-h8 

-i-3a6— lOx 

-8a2 

+  2.r.y— 9 

-f-3a6— 6.r 

+  10a2 

— 3av— 11 

-ah-'Hx 

-|-13a2 

+2-'.;^- 13 

+  2a6-i-7a; 

+  10^2       -{-12a^-6  -|-4a6+4x 


^2a^x  --.6a2+26  Gax^+S.r^ 

4-  «v/-^  4- 2a-— 36  _2aa'2— 6x4- 

_3a^a-  ~5a2— 86  -f3ax-— 10^4 

-V^a^/x  -\-4a''-2b  ^•:ax'-\-Sxf 

—\a^x  _3a^4-96  J^ax'-^-Wxl 


—  a>/a;  -  8a' — 26  -}-aa;2-f3a-J 


ADDITION.  n 


CASE  III. 

tfhen  the  Quaniiiies  are  unlike;  or  some  like  and  others  un^ 
like, 

RULE. 

Collect  all  the  like  quantities  together,  by  taking  theip 
sums  or  differences,  as  in  the  foregoing  cases,  and  set  down 
those  that  are  unlike,  one  after  another,  with  their  proper 
signs, 

EXAMPLES. 


5\y 

2xy-—2x^ 

2aa:— 30 

4aas 

Sx^-hxy 

3x2— 2aa; 

-xy 

x^+xy 

5x2— 3x^ 

■4:ax 

4x^ — 3xy 

3v'^H-lO 

4xy  6x2-f  xy  Sx^ — 20 


— ax^  la  X  — 5xy  —  b^-{-2a'^x- 

-fSflx^  dxy—Qax  bO  •\-2a^x 

—  0x2  2a^x^-\-  xy  a^x^-j- 1 20 


— 2ax2      10aV-i-5xi/-~ax     962-f  SaV+a^x-f  17< 


-i-da^y  2^x  -  I7y  2a?—3a^x 

— 2x?/2  3yx2/+lOx  x2-2aM" 

—3^2^  2x^y    +25y  Ba^'-lSxy 

— 8x^y  12x?/  — ^xy  xy  -{-32a- 

■^2xf  —Si/  4-18x2  20— 65x= 


Cl9^x-{-l2xy) 
{  'h2x'y-^2y/x  } 
f  2,-1-1  Ox— 1/       ) 


3a2|/— Si/^x— Sx-i/    ^  19yx  4-j2x^)  37a2— 3a-v/x— 12.c 

-  - 

y — 6  4x" —  2a  2  X  -  -J- 

20 


12  SUBTRACTION. 

EXAMPLES  FOR  PRACTICE. 

1.  Required  the  sum  of  i  (a+M  a"d  }  {^ — ^)=     Ans.  a. 

2.  Add  5.T  —  3a  +  6+7  and  —  4a— 3a;  -}-  26  —  9  toge- 
ther. Ans.  2x  —  7a  +  36  —  2. 

3.  Add  2a  +  36  -—  4c  ~  9  and  5a  —  36  4-  2c  -  lu  to- 
gether. Ans.  7a  —  2c  —  19. 

4.  Add  3a  -f  26  ~  5,  a  -f  56  —  c,  and  6a  —  2c  -{-  3  to- 
gether. Ans.  10a  4- 76  —  3c  —  2. 

5.  Add  x^  +  aa;2  +  6j;  -}-  2  and  x^  -{-  cx^  -f  t/x  —  1  to- 
gether. Ans.  2:r3  +  (o+o)  a;^  .|.  (6  + J)  a;  4"  1- 

6.  Add  6a:?/-12j^  -  4x^-1- 3.xt/,  4x2—2x7/,  and  -3x1/4- 
4 x^  together.  Ans.  4x»/ — 8x^ 

7.  Add 4ax-  130+3x2,  Sx^+Sax+Ox^,  7xi/— 4x2 4-90, 
and  v^a;4-40 — Qx^  together.  Ans.  7ax4-8x^4-7xy. 

8.  Add  2a2  — 3a64-26^— 3a^  36"— 2a=^4-a^— 56^,  4c''— 
S6H5a64-100,  and  20a64-16a2— 6c— 20  together. 

Ans.  13a24-2  a64-36Ha'— c'— 20— 6c._ 

^    .,,5a     3c2     7^6c     ^o64-x^      ,  8a     7c2     -.oa/*' 

9.  Add-7 V-' 9( — }-)and-^4 12^:^— 

6       a  X  ^     a     *         0       a  x 

'f6("-y:^)  together. 

Ans.  -^4 ^— 3(— -J-). 

b         a  X  d     ' 

10.  Add  3a24-46c— e24-10,  — 5a24-66c4-2e2— 15,  and 
— 4a'— 96c— 10e24-21  together. 

Ans.  6c -6a2— 96^4- 16. 

SUBTRACTION. 

Subtraction  is  the  taking  of  one  quantify  from  an- 
other ;  or  the  method  of  finding  the  difference  between  any 
two  quantities  of  the  same  kind ;  which  is  performed  as 
follows*  : 

*  This  rule  being  the  reverse  of  addition,  the  method  of  operation  must  be 
so  likewise.  It  depends  upon  this  principle,  that  to  subtract  an  affirmative 
quantity  from  an  afRrmativcj  i»  the  same  as  to  add  a  negative  quantity  lo  an 
affirmative. 


ttUBTKACTIUN.  lo 


RULE. 


Change  all  the  signs  (  -f  and  —  )  of  the  lower  line, 
\ir  quantities  that  are  to  be  subtracted,  into  the  contrary 
signs,  or  rather  conceive  them  to  be  so  changed,  and  then 
collect  the  terms  together,  as  in  the  several  cases  of  addi- 
tion. 

Examples. 

5a2— 26  x^—2y-{'S  5xy-\-Sx-'2 

2a^-\-5b  4x''+9y'~5  Sjcy—Sx-^7 


3a2-76  -3a;^-.lli/-|-8  2xr/-hl6x-f5 


Bxy-lS  Qy^^2y-6  lOSv^3xij 

-at/-}- 12  -2/2+3J/+2  -'X+S^xy 

6xy^30  Qf^iy^'j  7^7x—2xy 


-bx^y—Sa  4y/ax^2x"y  bx'^+y/x^4y 

-{-Sx^y  —  lb  S^ax-dxy^  Hx""  —  Hx^x^- 


■^8x^y^8a+7b     yax-2x^y-^5xy''  ■'X^+8x-i'2^x-'4y 


EXAMPLES  FOn  PRACTICE. 

i.  Find  the  difference  of  i  (a+b)  and  i  («  —  6).  Ans.  0, 

2.  From  3x-2a-6-f7,  take  8^3b+a+4x. 

Ans.  26  — x  — 3a— J. 

3.  From  3a+6+c-  2c/,  take  6~8c4-2fi— 8. 

Ans.  3aH-9c-4c/'HS. 


Thus,  according  to  Laplace,  we  can  write 

a  =  a+h—h (1), 

« — cc=  a—c-{'b—b (2)  ; 

■w  that  if  from  a  we  are  to  subtract  -j-  h,  or  —  h  ;  or,  what  amounts  to  the 
same  thing,  if  in  a  we  suppress  -f-^i  or — 6  ;  the  remainder  from  transforma- 
tion (1),  must  be  a — h  m  the  first  case,  and  rt-f-6  in  the  second.  Also,  i[ 
from  a—c  we  take  away  '\-b,  or  —6,  the  remainder,  from  (2),  will  be  a— c 
— ^,  or  a— c-f.6.  Ed. 

C 


14  MULTIPLICATION, 

4.  From  \2z'^'-2ax-9b\  take  Sx^— 7aa;-K 

Ans.  Sx^A-^ax-^-lOl- 

5.  From  20ax  —  5^:c  -f-  3a  take  iax-jr^x^  —a. 

Ans.  16ax— l0y^a;+4a, 
5.  From  5a6+262-c-f  6c-^>,  take  6^-  2ab  +  bc. 

Ans.  7a6-|-62-c  — 6, 
Q.  From  ax^'^hx^-\-cx'-'d,  take  fcx^+e^  — 2</. 

Ans.  a.T^-26a;-{-(c  — e)x+c/o 
3.  From-6ti-4&-  12c-}- 13r,  take  4x--.9a4-46  — 6c. 

Ans.  3a4-9a;— 86  — 7c- 

9.  From  6x"ySy'xy'-Qay,  take  2x''y-\-S\xyy-^4ay, 

Ans.  3a;^?/  — Cv'ar?/— 2ai/, 

10.  From  the  sum  o^4ax  —  150  -f  4a;^  5x^  +  Sax  -f 
lOx^,  and  90  —  2ax  —  12  ^  a; ;  take  the  sum  of  2ax  — 

30  -f-  7x2,  7^4  _  g^^  _  ^Q^    jjj^j   30  —  4  ^T  —  2x  -f- 
laV.  Ans.  llaa:4-60-x-  — 4aV. 

MULTIPLICATION. 

Multiplication,  or  the  finding  of  the  product  of  two 
or  more  quantities,  is  performed  in  the  same  manner  as  in 
arithmetic  ;  except  that  it  is  usual,  in  this  case,  to  begin 
the  operation  at  the  left  hand,  and  to  proceed  towards  the 
right,  or  contrary  to  the  way  of  multiplying  numbers. 

The  rule  is  commonly  divided  into  three  cases;  in  each 
of  which,  it  is  necessary  to  observe,  that  like  signs,  in  mul- 
tiplying, produce  +5  and  imlike  signs,  — . 

It  is  likewise  to  be  remarked,  that  powers,  or  roots  of 
the  same  quantity,  are  multiplied  together  by  adding  their 
indices  :  thus, 


L         JL 


=J. 


a  X  a-,  or  a^  X  (r  =  a^ ;  a^  X  a^  =  o^ ;  a^"  X  a^  =0® ;  and 

The  multiplication  of  compound  quantities,  is  also, 
sometimes,  barely  denoted  by  writing  them  down,  with 
their  proper  signs,  under  a  vinculum,  without  performing 
the  whole  operatioii,  as 


MULTIPLICATION.  U 

Sab  (a-  h),  or  2a^"+b\ 
Which  method  is  often  preferable  to  that  of  executing  the 
entire  process,  particularly  when  the  product  of  two  or 
more  factors  is  to  be  divided  by  some  other  quantity,  be- 
cause, in  this  case,  any  quantity  that  is  common  to  both  the 
divisor  and  dividend,  may  be  more  readily  suppressed  ;  as 
will  be  evident  from  various  instances  in  the  following  part 
of  the  work*. 

CASE  L 

When  the  factors  are  both  simple  quantities. 

RULE. 

Multiply  the  coefficients  of  the  two  terms  together,  ana 
to  the  product  annex  all  the  letters,  or  their  powers,  be- 
longing to  each,  after  the  manner  of  a  word  ;  and  the  re- 
sult, with  the  proper  sign  prefixed,  will  be  the  product  re 
quiredj. 

*  The  above  rule  for  the  signs  may  be  proved  thus  :  If  b,  b,  be  any  two 
quantities,  of  which  b  is  the  greater,'and  b — b  is  to  be  muhiplied  by  a,  it  is 
plain  that  the  product,  in  this  case,  must  be  less  than  as,  because  b — b  is  less 
than  B  ;  and,  consequently,  when  each  of  the  terms  of  the  former  are  multi- 
plied  by  a,  as  above,  tlie  result  will  be 

(b—L)  Xa  =  aB—ab. 

For  if  it  were  aB-fa6,  the  product  would  be  greater  than  as,  which  is  ab- 
surd. 

Also,  if  B  be  greater  than  b,  and  a  greater  than  a,  and  it  is  required  ic 
multiply  B— 6  by  a — a,  the  result  will  be 

(B— 6)  X  (a— «)  =  AB— OB— 6a-[-  ab. 

For  the  product  of  b—^/  by  a  is  a  (b— &),  or  ab— a6,  and  that  of  b— 6  by 
— rt,  which  is  to  be  taken  from  the  former,  is  — »  (b — b)  as  has  been  already 
shown  ;  whence  B—b  being  less  than  b,  it  is  evident  that  the  part  which  is  to 
be  taken  away  must  be  less  than  as ;  and  consequently  since  the  first  part 
of  this  product  is  — cm,  the  second  part  must  be  +  ab ;  for  if  it  were  — ab,  a 
greater  part  than  «b  would  be  to  be  taken  from  a  (b— t),  which  is  absurd. 

f  When  any  number  of  quantities  are  to  be  multiplied  together,  it  is  the 
same  thing  in  whatever  order  they  are  placed  :  thus,  if  ab  's  to  be  multiplied 
by  C,  the  product  is  either  abc,  acb,  or  bca,  «fcc.  ;  though  it  is  usual,  in  this 
case,  as  well  as  in  addition,  and  subtraction,  to  put  them  according  to  their 
rank  in  the  alphabet.  It  may  here  also  be  observed  in  conformity  to  the  ruk 
given  above  for  the  signs,  that  (+a)X(4-i),  or  T— a)  V  (— W  =;  +  a^ 


1 6  BIULTIPLIC  ATION. 


12a 
i5b 

E 

~2a 
+46 

XAM 

fors 

RU] 

PLES. 

+5a 
-Gar 

— 9x- 

— 56z 

S6ab 

-8a6 

— SOrtx 

+46fc.x'' 

lab 
— 5ar 

+  6a 

—  2X7/ 

—  7axy 
+  6a^ 

-S5a%c 

—SOa'x' 

+2xy 

+ay^z 
^Qaxy-^z- 

II. 

u  a  compow 

LE. 

— 42a2a;^- 

Sa^b 
2ba^ 

\2a^x 

—  2X2  7/ 

—  0^X2, 

+2x1/^ 

Ga'li' 

-24.a^x^y 

— 2aV?/' 

When  one  ofihefaci 

uZ  quantity. 

Multiply  every  term  of  the  compound  factor,  consider- 
ed as  a  multiplicand,  separately,  by  the  multiplier,  as  in  the 
former  case  ;  then  these  products,  placed  one  after  another, 
with  their  proper  signs,  will  be  the  whole  product  requir- 
ed. 

EXAMPLES. 

3a-2^'  6x7/- 8  a2-.2x  +  ! 

4a  3x  4x 


12a2-8a6  18x2v-24x  4/i2x— 8x^+4; 


BULTIPLICATIOJS.  i: 


12x  — a6 
5a 

35x'-7rz 
-2x 

3^/2 -hy- 2 
xy 

60ax-6a^b 

— 70x2  +  1 4ax- 

3xy^-\-xy"-2xp 

13x^-a^b 
—2a 

25az/4-3a3 
13a:2 

3x2-x3/+2i/2 
5x2 

-26aa;2_2a36 

325x3?/ -f39a2a;2 

15x'— 5x'y--10y 

CASE  llf. 

When  both  the  factors  are  compound  quantities^ 

RULE. 

Multiply  every  term  of  the  multiplicand  separately,  by 
each  term  of  the  multiplier,  setting  down  tlie  products  one 
after  another,  with  their  proper  signs  ;  then  add  the  several 
Jines  of  products  together,  and  their  sum  will  be  the  whole 
product  required. 

EXAMPLES. 


x-^y 
x-hy 

5x  +42/ 
3x-22/ 

J  5x2  +  1 22-2/ 

-10x^-81/2 

J  5x2+  2xy-Qy^ 

X2+2/ 
a-2+2/ 

x^+x^y 
+X22/+2/2 

x^+?x2  2/+2/2 

X3+X2/— 2/' 
X  -2/ 

x^-rxy 
-rxyi-y^ 

x3+x2  2/— ''*^2/^ 

— .x2  7/-.Xy2+2/' 

x2 +2x2/4-2/- 

x+y 
x-y 

x3      *      -2x2/2+2/3 

X2+X2/+3/2 
X   -y 

x^-{-xy 

~xt/-2/' 

rv'3+x22/+x2/2 
—  x-y  —  xy^'^y^ 

x2    *    -2/2 

x^       *  *   —2/^ 

c8 


16  MULTIPLICATION. 

EXAMPLES  FOR  PRACTICE 

1.  Required  the  product  of  .x^  ^xy-\-y^  and  x-\-y, 

Ans.  x^-\-if'. 

2.  Required    the  product  of  x^ -f-  x^y  -{-  ^y^  +  */  and 
x  —  y.  Ans.  x^— 2/*- 

3.  Required  the  product  of  x^-^-xy-^-f  and  x^ —xy-^ 
y^,  Ans.  x"'+a*2y^+7/'*. 

Q.  Required  the  product  of  32:2-2x?/-i-5,   and  x^-\-2xy 
— "B."  Ans.  3x*4-4T^?/-4xy-4x--f-16ri/~15. 

5.  Required  the    product  of  2a^— 3arc-h4a;^  and  5a^  — 
6ax-.2x-.  Ans.  10a*-27a=^i-|-34a2x-2_  ISax^- 8x*. 

6.  Required  the    product  of  bx'^-\-^ax^-\-^arx-\-n?,  and 
2x2— 3ax+a2.  Ans.  iOx^— Tax*— aV— 3a=^x^+a^ 

7.  Required   the  product  of  3x=^4-2xY-f3i/'   and  2r 
— 3xY+5i/=^. 

Ans.  6x^—5x5^/2— 6xy+21xy4-a:Y+  15x^ 

8.  Required   the  product  of  tx^—<ix-4-6x-c   and   x^  — 
<?x4-^-      Ans.  x^'-ax^  —  dx'^-\-{h-\-ad-\-e)x^—{c']rhd-\-ae). 

x'^-\-{cd\'€h)x — ce. 

9.  *  Required  the  product  of  the  four  following  factors, 
viz. 

I.  XL  III.  IV. 

{a-{-h)  (a2+a6+fe2)  (fl-6)  and  (a^-afe+i"). 

Ans.  a^ —h^. 

10.  Required  the   product  of  a^-j-Sa^x-j-Sax^+x^   and 
a^  —  Sa^x  -|-3ax- — x^. 

Ans.  a^— 3aV4-3a2x*— x^ 

11.  Required  the  product  of  a^+ct'c^+c^  and  a^— c^. 

Ans.  a^  —  c°, 

12.  Required   the   product   of  a--{-6^4-c^  — Q^  —  ac  — (" 
and  a  -}-6+c.  Ans.  a=^  -  3«6c+  6H-c'. 


*  I  would  advise  the  learner  to  perform  the  calculation  of  this  exannple 
several  ways,  viz.  First,  by  multiplying  the  product  of  the  factors  I.  anci 
II.  by  the  product  of  the  lactors  III.  and  IV.  Secondly,  by  multiplying  the 
product  of  the  factors  I.  and  III.  by  the  product  of  the  factors  II.  and  IV 
Thirdly,  by  multiplying  the  product  of  the  factors  I.  and  IV.  bv  the  product 
of  the  factors  II.  and  III.  The  last  method  is  the  most  concise';  See  EuleT' 
Algebra,  page  119.  Vol.  I.  Er. 


DIVISION.  1^ 


DIVISION. 

Division  is  the  converse  of  multiplication,  and  is  pei' 
formed  like  that  of  numbers  ;  the  rule  being  usually  di- 
vided into  three  cases  ;  in  each  of  which  like  signs  give 
4-  in  the  quotient,  and  unlike  signs  — ,  as  in  tinding  their 
products*. 

It  is  here  also  to  be  observed,  that  powers  and  roots  of 
the  same  quantity,  are  divided  by  subtracting  the  index  of 
the  divisor  from  that  of  the  dividend. 

Thus,  a^-r-a^,  or  —  =a  ;  a  "-7-«^,  or,  ~^=a'^ ; 

a*  a^ 

3 

3        2        a*         1  «*" 

a* -f.a3  or  — =a»  ^  .  and  a^-^rt^  or  — =a'"'' 
a3  a 

CASE  I. 

When  the  divisor  and  dividend  are  both  simple  compounds. 

JIULE. 

Set  the  dividend  over  the  divisor,  in  the  manner  of  a 
fraction,  and  reduce  it  to  its  simplest  form,  by  cancelling 
the  letters  and  figures  that  are  common  to  each  term. 

EXAMPLES. 

6a&-f-2a,  or  ^=36  ;  and  \2ax^^Sx,  or  ^-^^^4ax  ; 
2a  3x 

a      ,  ,  a 

a-7-«,  or-=  I  ;  and  a-^ a  or  —  =  —  I. 

a  —a 


*  According  to  the  rule  here  g;iven  for  sign?,  it  follows  that 
U-ab  ,   ^   — ab  — nh  -j-«6 

-f-6         '         — h  4.6  —b 

as  will  readily  appear  by  multipliplying  the  quotient  by  the  divisor;  the  sign? 
of  the  product  being  then  the  same  as  would  take  place  in  the  fornaer  rule  ; 


-iu  DIVISION. 


Also  -2a  -r-  3a,  or-^-— =—1 ;  and  9x^^Sx*=Sx'\ 


2a 
3a 

1.  Divide  I6x^  by  8.t,  and  12aV  by  — Sa^x. 

Ans.  2a-,  and  —  -—. 

2.  Divide  —  1 5ay^  by  Sa^j  and  —  1  Sax^y  by  —  Sax. 

Ans.  — 5y,  and  — p  . 


4 


2  ^       11  i  3Ji 

^>   Divide  — .-ra"  bv  -u" .  and  aar^  by  —  -a'^x*. 

3  •  5  '5' 


5  i     '- 
Ans.  —31^,  and  — -a^a;"*^^ 


3 


2 


4.  Dividel2a-Z>2by-3a26,  and  —  15ai/3  by  — 3a2/2.   ^ 

Ans.  —  46,  and  6^'*^. 

5.  Divide  —  ISa^a"  by   Sax^,  and  21  aVa:^  by  —  7a 
J.  1 

c^a-*.  Ans.  —3a,  and  —  Sax*. 

J  J.     J  — 

6.  Divide—  17a;=a^c  by — ox^ah^,  and  240-?/  by  By/xy, 

.         17a;6ac'8         ,  „    .— 

Ans. ,  and  Sa/xv- 

5 

CASE.  II. 

H'A^i  the  -diviscr  is  a  simple  quantity,  and  the  dividend  a 
compound  one. 

Divide  each  term  of  the  dividend  by  the  divisor,  as  ii. 
fhe  former  case  ;  setting  down  such  as  will  not  divide  in 
the  simplest  from  they  will  admit  of. 

EXAMPLES. 

.5+63)^2,,,  or  "-^=la+Ji=«-+^ 

lOaJ— loaa:)-r5a;  or  - — =26 — 2x. 

Oct 


DIVISION.  21 

SOax  — 48x2 
(30ax-48x=)  -f-  6ar,  or =  5a-.§x. 

1.  Let  Sx^-fex^+Sflx  — 15x  be  divided  by  3x. 

Ans.  x^+2x+a^B, 

2.  Let  3abc-{-12abx-9a%  be  divided  by  'Sab, 

Ans.  c-{-4a:  — 3a. 

3.  Let40a=^6=*-{-60a26='-l7«6  be  divided  by  -a6. 

Ans.  40a262— 60«64-l7. 

4.  Let  iSa^ic — \2acx^'\-5a(Phe  divided  by  — 5ac. 

12x2      d- 

Ans.  — 3a6H . 

5  c 

5.  Let  20ax^-fl5ax--f  10ax+6a  be  divided  by  5a. 

Ans.  4x34-3r2-f2x+l. 

6.  Let  ebcdzX4bzd^-'2b''2^  be  divided  by  2bz. 

Ans.  3c(i  4-2^2 -fes'. 

7.  Let  14a2  — 7afe+21ax  —28a  be  divided  by  7a. 

Ans.  2a — 64-3x— 4- 

8.  Let  -^Wab-\-60aP—l2a^b^  be  divided  by  — 4a6. 

Ans.  5"-1562-f3a6. 

CASE  IIL 

When  the  divisor  and  dividend  are  both  compound 
quantities. 

RULE. 

Set  them  down  in  the  same  manner  as  in  division  of 
numbers,  ranging  the  terms  of  each  of  them  so,  that  the 
higher  power  of  one  of  the  letters  may  stand  before  the 
lower. 

Then  divide  the  first  term  of  the  dividend  by  the  first 
term  of  the  divisor,  and  set  the  result  in  the  quotient, 
with  its  proper  sign,  or  simply  by  itself,  if  it  be  affirma- 
live. 


2'i  mvisioM. 

Tills  being  (lone,  multiply  the  whole  divisor  by  the 
term  thus  found  ;  and,  having  subtracted  the  result  from 
the  dividend,  bring  down  as  many  terms  to  the  rcmaindei 
as  are  rofjiiisile  for  the  next  operation,  which  perform  a.s 
before  ;  and  so  on,  till  the  work  is  finished,  as  in  common 
nrilhmclic. 


EXAMPLES. 


X"  -\-xy 

xy+y* 
xij+y'-' 


a-\'x)a^'{-5a-x-\-5ax^'\-x^{a'-\-^ax-\-x^- 


4a''x-h4ax^ 


ax^-i-x^ 


— G.r3+27x 


9r-27 

9X—21 


DIVISION.  23 


NoTK  1.  It'  tho  divisor  be  not  exactly  contained  in  the 
dividend,  tho  quantity  that  remains  alter  the  division  is 
finished,  must  bo  placed  over  the  divisor,  at  the  end  of  tlio 
'inoticnt,  in  the  form  of  a  fraction  ;  thus,"^ 


2x- 

a-^-x 


—  a^x  —  ax'^ 
ax--{-x^ 


^  In  (he  case  here  given,  the  operation  of  division  mny  be  considered  as 
terminated,  when  the  highest  power  of  the  letter,  in  tlie  first  or  leading  term 
of  the  remainder,  by  which  the  process  is  regulated,  is  le.«s  than  (he  power 
of  the  first  term  of  the  divisor  ;  or  wlien  the  tli  st  term  of  the  divisor  is  not 
contained  in  the  first  term  of  the  remainder  ;  as  the  succeeding  part  of  the 
quotient,  after  this,  instead  of  being  integral,  as  it  ought  to  be,  would  neces- 
sarily become  fractional. 


24  mvisioN. 

xi-y)x^  -i-y^  {x^  —x-y-^-uy*  -  y'^ 


Hy* 


x-^-y 


x*+x^y 

-x'y+y' 

—  x^y  —  x^y^ 

x^y^-k-y^ 
x^y^-\-xy'^ 

-xy^ 

+2/^ 

-r 

^.  Tho  (livisioQ  oi  quaDtities  may  also  be  sometiincd  caj- 
lied  on,  ad  infinitum,  like  a  decimal  fraction  ;  in  which  case 
a  few  of  ihe  leading  terms  of  the  quotient  will  generally  be 
.•sufficient  to  indicate  the  rest,  without  its  being  necessary 
to  continue  the  operation  ;  thus, 

a+x)a  ....  (1--+ ^-?-4-^  &c.* 
— x 


a 


x^ 


*  Now,  it  is  easy  to  perceive  (hat  Iho  next  or  6th  term  of  the  quotient  will 
*'  ~  r5'  ""^  ^^^  seventh  lenn  --  and  so  on,  alternately  j'^us  and  minus- 


DIVISION.  25 

And  by  a  process  similar  to  the  above,  it  may  be  shewn 
(hat 

=1+-- f— +-  +_,+_-|-A2C. 

a — X  a      (H       a*       a^       a' 

Whcie  the  law,  by  which  either  of  these  series  may  bo 

continued  at  pleasure,  is  obvious*. 


ihis  is  called  the  law  of  continuation  of  the  series.    And  the  sum  of  all  the 

lerms  when  infinitely  continued  is  said  to  be  equal  to  the  fraction  , 

2 
Thus  we  scythe  vulvar  fraction-  when  rcducedtoadeci:nalis  =  22222,  <fec. 

Mifjnitely  continued.  The  terms  in  the  quotient  are  found  by  dividing  the  re« 
snainders  by  a,  the  first  term  of  the  divisor  ;  thus,  the  first  remainder — x  di- 
vided by  «,  gives the  second  term  in  the  quotient;  and  the  second  re* 

nainder  J divided  by  o  gives  •+•  —  the  third  term,  &c 

»       rt  •'"'(12  ' 

*  In  tljis  example,  if  x  be  less  than  a,  the  series  is  convergent,  or  the  value 
af  (he  terms  continually  diminish ;  but  when  x  is  greater  than  a,  it,  is  said  to 
diverge. 

To  explain  this  by  numbers  :  suppose  a  =.'},  and  x  =2, 

Then,  1  +5+^-1-^:1,  &c 
The  corresponding  values  are, 

where  the  fractions  or  terms  ot  the  series  grow  less  and  less,  and  the  farther 
they  are  extended,  the  n»ore  they  converge  or  approximate  to  0,  which  is 
supposed  to  be  the  last  term  or  limit. 
IJutifa=2,  ai»da;  =  3, 

Thenl+^4--  +  -.&c- 
'   a      (12     •  a3  ' 

The  corresponding  values  are, 

3      9      27 

In  which  the  terms  become  larger  and  larger.  This  is  calied  a  divcrffin*' 
ieries.  ^    *^ 

I  f  a;  =»  1 ,  and  a  =al  ;„  the  preceding  example  : 

Vow,  because  y— rra.i.,  it  has  been  said  that  1—1+1—1,  &c,  infinitely 

f^ontinued,  is  «=  -J- :  a  singular  conclusion,  when  it  is  perceived  from  the  terms 
"hrmselvcs,  that  their  sum  must  necessarily  be  either  0  or-l-l,  to  whatever 


26  DIVISION, 


EXAMPLES  FOR  PRACTICE. 

1.  Let  (^ — ^ax'\-x^  be  divided  by  o  — a:.  Ans.  a— s: 

2.  Let  x^ — Sax^+Sa^x — a^  be  divided  by  x — a. 

Ans.  x^ — '2ax'\-(j?> 

3.  Let  a^-|-5a^x+5a2"-}-a;^  be  divided  by  a-\-x. 

Ans.  c^-\-\ai'\-x'. 

4.  Let  22/3— 192/2-1-26?/— 16  be  divided  by  i/— 8. 

Ans.  22/2_3^+2. 

5.  Let  x^-f-l  be  divided  by  x-(-1j  and  x^ — 1  by  x — 1. 

Ans.  rr'— x^-l-x^— x+l,  and  x^-\-x''-\-x''-\-x'^x-\-\. 

6.  Let   48x^— 76ax=^+64a2x-Hl05a=^  be  divided  by  2x 
—3a.  Ans.  24x2— 2ax— 35a^ 

7.  Let  4x*— 9x2+6x-  1  be  divided  by  2x2-j-3x— I. 

Ans  2x2— 3x+  J 

8.  Let  x^ — aV^2a3x— a^  be  divided  by  x^— ax-j-a^. 

Ans.  x^-f  ax— a^ 

9.  Let  6x*— 96  be  divided  by  3x~6,  and  aP-\-x^  by  a 

Ans.  2x3-f4x2-}-Sx-t-16,  and  a^— a^x-f a^x^— ax^— x^ 

10.  Let  32x^+243  be  divided  by  2x+3,   and  x^— a«  by 
X — a. 

Ans.  16x^— 24x^-1- 36x-—54x+81,  and  x^-f-o^^+aV 
-f-a^x^-f-a^x-fa^ 

11.  Let  6*— 32/'  be  divided  by  6—7/,  and  a*4-4a26-l-86^ 
by  a+26. 

Ans.  63+62t/-|-ft/-{-i,'^— ^,  and  a^— 2a26-|-4a^-^ 

a-}-26 


extent  ihe  division  is  supposed  to  be  continued.  The  real  question,  however, 
results  from  the  fractional  parts,  which  (by  the  division)  is  always  •\-~  when 
iJie  sum  of  the  terms  is  0,  and  — \  when  the  sum  is  -{-  1  :  consequenlly  \  if 
the  true  quotient  in  the  former  ca"se,  and  1— -l  in  the  other.  EiT. 


ALGEBRAIC  FRACTIONS.  27 

12.  Let  x^-^-px-^q  be  divided  by  x-f-a,  and  x^ — px^-{^ 
fc — r  by  X — a. 

Ans.  x-\-p — a-\ , ,  and    x^-f-   (a — p)   x-^ap 

^  x-\-a  V       i  / 

a^ — a^p-\-aq — r 
X — a 

13.  Letl— Sx-f  ux2— i0a;3-f-5a;*-x5  b© divided  by  1— 
^x-^-x^  Ans.  I  -3x 4-3x2 -a;3. 

14.  Let  a*-f  46^  b«^  divided  by  a^— •2«6+262. 

Ans.  a2-f-2fi64-262. 

15.  a^ — 5a*a;+IOa'r2 — lOi^rc'-fSaa;^— x^  be  divided  by 
^2— 2aT4-a:-.  Ans.  a'— 3a2.r-{-3ax2— a;=^. 

16.  Let  a^+fi.^  be  divided  by  a^-{-ab^/2-\-bK 

Ans.  a^ — ab^2-\-b^l 

OF  ALGEBRAIC  FRACTIONS. 

Algebraic  fractions  have  the  same  names  and  rules  of 
operation  as  numeral  fractions  in  common  arithmetic ;  and 
the  methods  of  redu<-ing  them,  in  either  of  these  branches^ 
to  their  most  convenient  forms,  are  as  follows  : 

CASE  L 

To  find  the  greatest  common  measure  of  the  terms  of  afrac^ 
*  tion. 


RULE. 

1.  Arrange  the  two  quantities  according  to  the  order  of 
their  powers,  and  divide  that  which  is  of  the  highest  di- 
mensions by  thp  other,  having  first  expunged  any  factor, 
that  may  be  contained  in  all  the  ttrrns  of  the  divisor,  with- 
out being  common  to  those  of  the  dividend. 

2.  Divide  this  divisor  by  the  remainder,  simplified,  if 
necessary,  as  before ;  and  so  on,  for  each  successive  rQ<=' 
mainder  and  its  preceding  divisor,   till  nothing  remain^ 


2d 


ALGEBRAIC  FRACTIONS, 


when  the  divisor  last  used  will  be  the  greatest  commoi- 
measure  required  ;  and  if  such  a  divisor  cannot  be  found 
the  terms  of  the  fraction  have  no  comaion  measure.* 

Note.  If  any  of  the  divisors,  in  th^  course  of  the  ope- 
ration, become  neu^ative,  they  may  have  their  signs  chang- 
ed, or  be  taken  affirmatively,  without  altering  the  truth  of 
the  result ;  and  if  the  first  terra  of  a  divisor  should  not  be 
exactly  contained  in  the  first  term  of  the  dividend,  the  seve- 
ral terms*  of  the  latter  may  be  multiplied  by  any  number, 
or  quantity,  that  will  render  the  division  complete!. 

EXAMPLES. 

1.  Required  the  greatest  common  measure  of  the  fra: 
tion 


1^5  ^  jC'J  • 

x'^l)xl'^x\:c 
x^ — -x 

x^-i-x 
or  x'+  1 

:r4-2(V^2 
x'-hx' 

•* 

* 

*  If,  by  proceeding  in  thismajiner  no  conipour.d  divisor  can  be  found,  tha' 
19,  if  the  last  remainder  be  only  a  simple  quantity,  we  may  conclude  ihe  case 
proposed  does  not  admit  of  any,  but  is  already  in  its  lowest  terms.    Thus,  or 

■  c  r.    r       ■                     1            ,    u     a3^2a2x-^3ax2  -^4x3  _  , 
instance,  if  the  fraction  proposed  were  to  be — r ;iti:; 

plain  by  inspection,  that  it  is  not  reducible  by  any  simp  e  div'sor;  but  tc 
Koow  whether  it  may  t)ot,  by  a  compound  one,  I  proceed  as  above,  and  find 
the  last  remainder  to  be  the  simple  qu.ntity  7x2  :  whence  1  conclude  that 
the  fraction  is  already  in  its  lowest  terms. 

■f  In  finding  the  greatest  common  measure  of  two  quantities,  either  of  them 
may  be  multiplied,  or  divided,  by  any  quantity,  which  is  not  a  divisor  of  the 
ether,  or  that  contains  no  factor  which  is  common  to  them  both,  without  iu 
any  respect  changing  the  result. 

It  may  here  also  Ue  farther  added,  that  the  common  measure,  or  divisor. 
W any  number  of  quantities,  may  be  determined  in  a  similar  manner  to  that 
given  above,  by  first  finding  the  common  measure  of  two  of  them,  and  thers 
ef  that  coramon  measure  and  a  third ;  and  so  on  to  the  lest. 


ALGEBRAIC  FRACTIONS.  2& 

Whence  x^-{- 1  is  the  g^reatest  common  measure  required. 
2.  Required  the  greatest  common  measure  of  the  frac^ 

x'+2  x'-^h'x 


*^^0x''—2b^x 

or  x-l-6       I  x^'\-2bx'{-b\x'{'b 
rr"-f-6x 


* 
Where  x-\-b  is  the  greatest  common  measure  required. 
3.  Required  the  greatest  common  measure  of  the  fraC' 

•){Ojj 

3 


i2a'-Cm''—9a+3{4a 
I2a2— Sa^— 4a 


2a^— .5£i-f3)3a34.2««- 
2 


6a2-f-4a— 2(3 
6a2_l5a+9 


llct — 11  or  a — 1 
^Viiere,  since  a— l)2a2— 5./.-f  3(2a— -•^,  it  follows  that  the 
fast  divisor  a — i  is  the  common  measure  required. 

*  Here,  I  divide  the  remainder  — 26^2  — 2x2  x  by  — 2x6,  (its  greatest  sim- 
pte  divisor)  and  (he  quotient  is  x-^-b;  and  then  1  divide  the  last  divisor  bv 


^  ALGEBRAIC  FRACTIONS. 

In  which  case  the  common  process  has  been  interrupted 
in  the  last  step,  merely  to  prevent  the  work  overrunning 
the  page. 

4.  It  is  required  to  find  the  greatest  common  measure 

Ans.  x  — c. 

5,  Required  the  greatest  common  measure  of  the  frac- 

a*— a^ 

lion  — o-r—i'  -^"s.  a- — a-. 

a' —  a^x — (ix--\-  x-^ 

6.  Required  the  greatest  common  measure  of  the  frae- 

hon -j-^ r.  Ans.  a;-+aa;+a''. 

X  -\-  ax-  —  rrx — a- 

7,  Required  the  greatest  common  measure  of  the  frac 

7a-~2  v<6^-66- 
S.  *  Required  the  greatest  common  measure  of  the  frac- 

x^-rax^-\-bx'-^ — 2a-a;  +  6ax — 26a-  ,  ,  ^ 

hon 2— J — — — , .  Ans.  a-f  2a- 

x^—bx-r2ax — 2ab 


*  This  fraction  can  be  reduced  by  Simpson's  rule  (page  50)  thus  : 
Fractions  tliat  have  in  them  ffiore  than  two  different  letters,  and  one  of  the 
letters  rises  onlj  to  a  single  dimension,  either  in  the  numerator,  or  denomina 
lor,  it  will  be  best  to  divide  the  said  numerator  or  denominator  (whictiever 
it  is)  into  two  parts  so  that  the  said  letter  may  be  four.d  in  every  term  of  the 
one  part,  and  be  totally  excluded  out  of  the  other;  this  being  done,  let  thtr 
greatest  common  divisor  of  these  two  parts  be  found,  which  iviil  evidently 
be  a  divisor  to  the  whole,  and  by  which  the  division  of  the  quantity  is  to  be 
tried  ;  as  in  ths  following  exaniple,  where  the  fraction  given  ij 
a'S  -\-ax2 — bx2 — 2a2  x-\-  Lax — 2Lus 

x2 — hx^2ax — 2ab 
Here  the  denominator  being  the  least  compounded,  and  b  rising  therein  to  a 
single  dimension  only,  I  divide  tiie  same  into  the  parts  x2  J^2ax,  and  — bx~ 
2ab  ;  which,  by  inspection,  appear  to  be  equal  to  (x  ■i~2a)Xx,  and  (i-|-2<7- 
4-— 5.    Therefore  x-j-So  is  a  divisor  to  both  the  parts,  and  likewise  to  the 
whole,   expressed  by  Cx-\-2a)yi  ■.^—^'^''i    so  ihat  one  of  these  factors,   i; 
the  fraction  given  can  be  reduced  to  lower  terms,  must  also  measure  the  nu 
merator  :  but  the  former  will  be  found  to  succeed,  the  quotient  coming  ou 
s2 — ax-\-bx—ab,  exactly  ;  whence  the  fraction  itself  is  reduced  to 

.  ,  .    T. ,  which  is  not  reducible  farther  by  x — b,  since  the  divi 

X — b 
aion  does  not  terminate  without  a  remainder,  as  upon  trial  will  be  found. 

This  rule  is  sometimes  of  great  utility,  because  it  spares  great  labour,  anr 
li  Yery  expeditious  in  reducing  several  fractions.  £»- 


ALGEBRAIC  FRACTIONS.  M 

9.  Required  the  greatest  common  measure  of  the  frac- 
Hon -"^        .     Ans.  a:-+2ax  -  U\ 

10.  Required  the  greatest  common  measure  of  the  frac^ 
tion  -jrTn"n7^'rfi~nn~ri'  Ans.  x+b, 

11.  Required  the  greatest  common  measure  of  the  frac- 

CASE  II. 

To  reduce  fractions  to  their  lowest  or  most  simple  terms, 

RULE. 

Divide  the  terms  of  the  fraction  by  any  number,  or  quan- 
tity, that  will  divide  each  of  them  without  leaving  a  remain 
der  ;  or  find  their  greatest  common  measure,  as  in  the  las 
rule,  by  which  divide  both  the  numerator  and  denominator, 
and  it  will  give  the  fraction  required. 

EXAMPLKS, 

,     „    J        a^bc       ,      x^  ,    .    , 

1.  liCduce  ,  .,..,  and  — , — 5  to  their  lowest  terms, 
oa-o-        ax-jr^" 

Here  3-^„=-=TAns.  And  — - — 0==— r— ^  Ans, 
5a'*6^     56  ax-f-x'-     a-j-x 

2.  It  is  required  to  reduce  -^ — ; — r^-  to  its  lowest  terms > 
-  ^  a-c+a-x 


Here  cx-^x^ 
or     c-]-x 


a^c-x-a^x 


Wrience  c-f-x  is  the  greatest  common  measure  5 
and  c~f-.r)4--i — 7-=-^  the  fraction  required. 


3^  ALGEBRAIC  FRACTIONS. 

3.  It  is  required  to  reduce  -^  ,  „, — r-r,  to     its     lowes! 

terms. 

x^-^2bx'\-b^)x^'-b^x{x 


or  x+6  I  a:24.26a'-{-6\x+^ 


Whence  x-\-h  is  the  greatest  common  measure  :  and  x-f-b] 

x^  "^  b~x  x"  ^~  b  T 

~ — ; — rT7= — r-7-  the  fraction  required. 
:f'f-2bx-\'b-       a; +  6 

And  the  same  answer  would  have  been  found,  if  x^ — 6v 
had  been  made  the  divisor  instead  o^  x^-\-2bx-{-b^. 


4,  It  is  required  to  reduce  -^ t-t.  to  its  lowest  terms, 

^  x^  —  wx-^ 

Ans. ^ . 

o.  It  13  required  to  reduce  ^  .,  , ,  .^  .,  to  its  lowest 

^  6a-+llax  +  ox- 

3a  —  a- 
erme.  Ans.  - — ; — . 

Sa-\-x 

n     T    ■  J  ,         2:c3--16:r— 6^        .         , 

6.  It  IS  required  to  reduce  -— ^r — ^^ tr  to     its     lowest 

3a''—  24.r — 9 

*erms.  Ans,  #. 

^    ^    .  .J  ,  9a-^+2,r-I-4.r^-— x-i-l       " 

7.  It  13  required   to  reduce  , ,   ,    >.  ^  .  ,7—; rr;  to 

^  lox'^.- 2x^4- 10a:- — x+2 

3  lowest  terms.  Ans.  -^^j^^j^-^j 


ALGEBRAIC  FRACTIONS.  3& 

3,  It  is  required  to  reduce  — ^-^ r — — ^^,-r-^  to  its 

4a^d—  4acd— "ifac+Sc^ 

lowest  terms.  Ans.  — —7—. r--- 

4a  d — 2c^ 

CASE  Hi. 

To  reduce  a  mixed  quantity  to  an  improper  fraction. 

RULE. 

Multiply  the  integral  part  by  the  denominator  of  the 
fraction,  and  to  the  product  add  the  nu'nerator,  when  it  is 
affirmative,  or  subtract  it  when  nega  ive  ;  then  the  result^ 
placed  over  the  denominator,  will  give  the  improper  frac- 
tion required, 

EXAMPLES. 

1.  Reduce  3|  and  a  — -  to  improper  fractiofig, 

XT         00     3X54-2      16-f-2      17  , 

Here  3|=t= -—— =  -  Ans. 

5  5  5 

.     ,         h     aXc  —  b     ac—b  . 

And  a  --= = Ans. 

c  c  c 

a  q2  —  x^ 

2.  Reduce  xH — andx — to    improper     frac- 

X  X 

tions. 


Herea-f-= = — _  Ans. 

XX  X 

And  x* = =L Ans. 

XXX 


*  xXx  =ax2 .  la  adding-  the  numerator  a2—x2 ,  the  sign  --  affixed  to 
the  fraction 1,  denotes  that  the  whole  of 'hat  fraction  is  to  be  subtract- 
ed, and  consequently  that  the  signs  of  each  terra  of  the  numerator  must  be 


ALGEBRAIC  FRACTIONS. 


2x 

3.  Lei  1  —  —  be  reduced  to  an  improper  fraction. 

^  ,        a— 2x 

Ans. 

a 

O      I 

1.  Let  5a be  reduced  to  an  improper  fraction. 

^  .        5a2— 3i+t 

Ans. 

a 

o.  Let  a;  —  — — be  reduced  to  an  improper  fraction. 

'^^                                              2ax — a — x^ 
Ans. 


2a 

22-— .7 
6.  Let  5-{ — - —  be  reduced  to  an  improper  fraction. 

Ans.  -3^. 

X — a  —  1 

7,  Let  1 be  reduced  to  an  improper  fraction. 

«                                                      2a— a; +  1 
Ans. » 

a 

x  — 3 

8.  Let  \-\-2x —  be  reduced   to  an  improper  frac- 

ox 

r0x2+4.T-|-3 

:ion.  Ans. ;: 

bx 

CASE  IV. 

To  reduce  an  improper  friction  to  a  whole  or  mixed 
quantity, 

RUi.E. 

Divide  the  numerator  by  the  denominator,  for  the  inte- 
gral part,  and  place  the  rcnainder,  if  any,  over  the  deno- 
minator, for  the  fractional  part ;  then  the  two,  joined  to- 
gether, with  the  proper  sign  between  them,  will  give  the 
mixed  quantity  required. 

changed  when  if   is  combined  with  x^  ,  hence  the  improper  fraction  is 

Xo — a2+i2        2x2  _a2 

-= or .  Et). 

X  x 


ALGEBRAIC  FRACTIONS,  35 


EXAMPLES. 


„    ,        27         ax+a^ 

I,  Reduce   -  and      to  mixed  quantities. 

5  X 

27 
Here  _=27-r-o=6|  Ans; 

.     ,  aa-ha^             ,     ,,   .             i   ^^    » 
And— =(aa'+a^)-r-35=ai Ans. 

X  ^  X 


2.  It  is  required  to  reduce  the  fraction to  a 

whole  quantity.  Ans.  a  — ac^. 

ab  —  '^a^ 

3.  It  is  required  to  reduce  the  fraction r- —  to  a  mix- 

ad  quantity.  Ans.  1 — — , 

4.  It  is  required  to  reduce  the  fraction to   a  mix- 

a — X 

2x^ 

ed  quantity.  Ans.  a-^-x-] . 

^  a — a; 

5.  It  is  required  to  reduce  the  fraction ~  to  a  whole 

x-y 

quantity.  Ans.  x^-^-xy-^-y^. 

6.  It  is  required  to  reduce  the  fraction to  a 

mixed  quantity,  Ans.  2x  —  1  +t-. 

CASE  V. 

To  reduce  fractions  to  other  equivalent  ones,  that  shall  have 
a  common  denominator. 


RULE. 

Multiply  each  of  the  numerators,  separately,   into  all 
the  denominators,  except  its  own,  for  the  new  numerators. 


30  ALGEBRAIC  FRACTIOTSS. 

and  all  the  denominators  together  for  a  common  denom. 
nator*, 

EXAMPLES. 

1 .  Reduce  7-  and  -  to  fractions  that  shall  have  a  common 

6        c 

denominator. 

Here  aXc=^ac  }  ., 

^>^^_^2  J  the  new  numerators. 

bXc=bc  the  common  denominator. 

Whence,  -  and  --  =—  and  ;— ,  the  fractions  required. 

b  c      be  be  ^ 

2x  b 

2.  Reduce  —  and  -  to  equivalent  fractions  having  a  com- 

2cx       .  ah 
mon  denommator.  Ans.  —  and  — . 

ac  ac 

3.  Reduce  -  and to  equivalent  fractions    having    a 

common  denommator.  Ans.  t-  and  —. . 

be  be 

4.  Reduce  —   — ,  and  d,  to  equivalent  fractions  having  a 

J        2a'  So  g^^  4^^  g^^^ 

common  denominator.  Ans.  — ,  — ,  and . 

6ac  6ac  dac 

3  2x  4x 

5.  Reduce  -,-—  and  a-j— —  to  fractions  having  a  com- 

,  .     ,  «        45  40a;      ,  60a. -f  48.x 

mon  denominator.  Ans.  — , and . 

60   60  60 


*  It  may  here  be  remarked,  that  if  the  numerator  and  denominator  of  a 
fraction  be  either  both  multiplied,  or  both  divided,  by  the  same  number  or 
quantity,  its  value  will  not  be  altered  ;  thus 

2      2X36         ,3  2-r3      1         a       ac         ,   ab       a 

.  3=3X3="9' '"^  12  =12^3^4' ^^ft'^  6^'  ^"^ -ZF^c 
which  method  is  often  of  great  use  in  reducing  fractions  more  readily  (o  ? 
common  denominator. 


ALOEBRAIC  FRACTIONS.  37 

t>.  Reduce  -,  — ,  and ,  to  fractions  having  a  common 

2    7  a-—x 

,  .     ,  ,        7a2— 7ax  6ax— Sx^  ^  14a4-14x 

i^nominator.  Ans.  — —- ,— —-»  &i— t-t-  ^ 

14a— 14x  14a— 14x'      14a— 14?; 

CASE  VI. 

To  add  fractional  quantities  together. 

RULE. 

Reduce  the  fractions,  if  necessary,  to  a  common  deno- 
minator ;  then  add  all  the  numerators  together,  and  under 
their  surn  put  the  common  denominator,  and  it  will  give  the 
sum  of  the  fractions  required*. 

EXAMPLES. 


X  X 

i.  It  is  required  to  find  the  sum  of-  and  -. 

2  o 


Here    ^^  Jl"*  5   the  numerators. 


And  2X3=6  the  common  denominatoro 

^,^,  dx  .  2x      5a;     ,  .     , 

Whence  -—+—  =  -—,  the  simi  required. 
6       6       6 

2.  It  is  required  to  find  the  sum  of-,  -,  and^r- 

0    a        J 

HereaXrfX/=arf/, 


:j=adj) 
d=ebd  ) 


cX  LXf=cbfy  the  numerators. 
eX6X^ 


*  In  the  adding  or  subtracting  of  mixed  quantities,  it  is  best  to  bring  the 
fractional  parts  only  to  a  commoo  denominator,  and  then  to  affix  their  sum 
■or  difference  to  the  sum  or  difference  of  the  integral  parts,  interposing  tb*" 
proper  sign. 

E 


OS  ALGEBRAIC  FRACTIONS. 

And  bXdXf=bdf  the  common  denominator. 

.  adf  ,  cbf  ,  ebd     adf-^-cbf-^-ebd  ^ 

^^^°^^  bif^uf^bTr    w —  '^V"'"- 

3.  It  is  required  to  find  the  sum  of  a and  b-\ 

be 

Here,  taking  only  the  fractional  parts, 
We  shall  have  |  2alx6=2al  ^  ^^^  numerators. 

And  bXc=bc  the  common  denominators. 

3cx'  .  t  I  2062:         ,  ,  ,  2abx-^-3cx^  . 

Whence  a — r-'+b-i — -—  =a-|-6i ^r the  sum 

be  be  be 

4.  It  is  required  to  find  the  sum  of  ~-  and  — -. 

6  7 

Ans.  — . 

3x         X 

3,  It  is  required  to  find  the  sum  of  -—  and  -. 

^  2a        5 

-        l5z-\-2ax 
Ans. . 

IQa 

X    X         ,x 


6.  It  is  required  to  find  the  sum  of  -,  -,  and 


I3x 

Ans. . 

12 


7.  It  is  required  to" find  the  sum  of  ~  and  —r— . 

27a;~14 

2x  8x 

8.  Required  the  sum  of  2a,  3a-f— r-j  and  a—-^- 

22x 
Ans.  6a — --. 
4i> 

9.  Required  the  sum  of  2a-\—-y ,  and -. 

^  5>    a—x  a 

^     ,  «  ,  3a2a'-3ax-2--53:2 
Ans.  2a+2H ^t—^ 


ALGEBRAIC  FRACTIONS.  3^^ 

10.  Required  the  sum  of  5xH — — -  and  4a; — . 

Ans.  9a;i -~ * 

iSr 

U.  It  is  required  to  find  the  sum  of  5a;,  -     ,  and  —^ — ■■ 

da+3ax4-6a; 


Ans.  5x-\ 


CASE  VII. 
To  subtract  one  fractional  quantity  from  another. 


RULE. 

Reduce  the  fractions  to  a  common  denominator,  if  ne- 
cessary, as  in  addiuon ;  then  subtract  the  less  numerator 
from  the  greater,  and  under  the  diftV-ence  write  the  com* 
mon  denominator,  and  it  will  give  the  difference  of  the 
fractions  required. 

EXAMPLES. 

2a;         3x 
I ,  It  is  required  to  find  the  difference  of  -—  and  — -. 

3  5 


Here^^^g_  .'"'*  J  the  numeators. 


And  3X5—15  the  common  denominator. 

Whence  —= — tt^  ,"£:»  '^®  difference  required. 
15         1 5      15 

X  ■"  a 
2.  It  is  required  to  find  the   difference   of  -—t-  and 

26 

2a  — 4a; 


3c 

TT       (a;  — tt)X3c  =  3cx  — 3ac      >  ,, 

"^'^  (2o-4i)  X26=4«6  -84*  \  *«  ""meiators;- 


40  ALGEBRAIC  FRACTIONS 

And  26X3c=:66c  the  common  denominator. 

,_,            3cx— 3ac      4ab  -8bx       3cx — 3ac  —  4a64-8^i 
Whence  — -; — 


66c  66c  66c 

the  difierence  required. 

I2x         3x  4x 

3.  Required  the  difference  of-—  and  — .     Ans.  ^-hi^ 

4.  Required  the  difference  of  I5y  and  — --^. 

Ans.-|~ 

3.  Required  the  difference  of -r and^-; — . 

^axc 
Ans.^,--^ 

26a— '26x— c^' 


X  "^  CL  X 

6.  Required  the  difference  of  x and  x-f-rr» 

^  c  '2b 


Ans. 


26c 


7.  Required  the  difference  of  aH — ■ —  and  a . 

^  a-\-x  a — X 

.       2a24-2x" 
Ans. — 5 TT 

O'^  —  X" 

2x4-7 

8.  Required  the  difference  of  ax  + — - —  and  x  — 

5x~6  .  86x  — 99 

.  Ans.  ax  — < 

21  ^u&.  ax  jgg 

3x  -  5 

9.  Required  the   difference   of  2x-i — ,   and  3r-f 

ilx— 10  .  ,  32X+5 

' — .  Ans.  X- 


105 

a  -X 


10.  Required    the    difference    of  a  H — ; — :— r 
^  a(a-t-x) 

a-i-x  -  4x 

Ans.  a  — ■ 


and 


a(a  — x)'  *         aJ^"^ 


ALGEBRAIC  FRACTIONS.  41 

CASE  VIII. 
To  multiply  fractional  quantities  together, 

RULE. 

Multiply  the  numerators  together  for  a  new  numeratoi:- 
iind  the  denominators  for  a  new  denominator  ;  and  the  for- 
mer of  these  being  placed  over  the  latter,  will  give  the 
product  of  the  fractions,  as  required*. 

EXAMPLES. 

1.  It  is  required  to  find  the  product  of  -  and-^. 

6X9'~"54'~27        P*'°^^^*  required. 

2.  It  is  required  to  find  the  continued  product  pt 
■X  4a;        .  lOx 

„       xX4xX10a;      40^3     4x3 

Here  -— — ^    -—  -=■ =:rT-  the  product. 

2X5X21        210     21  ^ 

'3.  It  is  required  to  find  the  product  of  -  and . 

Here  ■—— 4=  -i the  product. 

aX(a  — x)     a^^ax 

4\  It  is  required  to  find  the  product  of  -^  and  — . 

Ans.-rr, 


»  When  the  numerator  of  one  of  the  fractions  to  be  nnultiplied,  and  the  de 
nominator  of  the  other,  can  be  divided  by  some  quantity  which  is  commojr. 
'0  each  of  them  the  quotients  may  be  used  instead  of  the  fractions  them 
sielves. 

Also,  when  a  fraction  is  to  be  multiplied  by  an  integer,  it  is  the  same 
thing  whether  the  numerator  be  multiplied  by  it,  or  the  denominator  divJd 
f,d  by  it.  Or  if  an  integer  is  to  be  multiplied  by  a  fraction,  or  a  fractoa  by 
an  integer,  the  integer  may  be  considered  as  having  unity  for  its  denoraijBa- 
tor,  and  the  two  be  thenfliultiplied  together  as  usual. 

£2 


42  ALGEBRAIC  FRACTIONS. 

5,  It  is  required  to  find  the  product  of  —  and  -— — . 

Ans.  -r— ». 
5ff 

6.  It  is  required  to  find  the    continued  product  of  -^, 

ix2  a  .  Sax" 

-y  and  —r—'  Ans. 


7  '  a+a;  21a-f  2a; 

7.  It  is    required   to    find   the  continued    product  o^ 

2x   Sab        ^  5ijc  ^  ^ 

— ,  —.and  --7-.  Ans.  \bax 

hx 

8.  It  is  required  to  find  the  product  of  2a-\ and  3a  — 

b  2h     b^ 

— .  Ans.  6a^-{-3bx r 

ax  X      a- 

9.  It  is  required  to  find  the   continued   product  of  3a', 
x+l        ,  x~l  ^       3a;3-3.x 
t; — »  and — —7-.                                              Ans.— -^rr-rT 

2a  a-f-6  2a^+2ab 

10.  It  is  required   to   find  the   continued    product   0 

a^— x^    a=  — 6^          ,      ,     ax  .       a^^a^h 

' — r-j-j ; — T)  and  air •  Ans. 

a-t-6    ax-\-x^                 a — x  x 

CASE  IX. 

To  divide  one  fractional  quantity  by  another,  ^ 

RULE. 

Multiply  the  denominator  of  the  divisor  by  the  nume 
rator  of  the  dividend,  for  the  numerator  ;  and  the  nume- 
rator of  the  divisor  by  the  denominator  of  the  dividend, 
for  the  denominator.  Or,  which  is  more  convenient  in 
practice,  multiply  the  dividend  by  the  reciprocal  of  the 
divisor,  and  the  product  will  be  the  quotient  required*. 

*  When  a  fraction  is  to  be  divided  by  an   integer,  it  is  (he  same  thin^ 
■whether  the  numerator  be  divided  by  it,  or  the  denomiBafor  multiplied  bj 


ALGEBRAIC  FRACTIONS.  43 


EXAMPLES. 

i .  It  is  required  to  divide  -  by  — . 

:?.  It  is  required  to  divide  —  by  — . 

__       ^a^,  d      2ad      ad    . 
Here  -j-  X— =—-=—-  Ans. 
6      4c     4oc      46c 

3.  It  is  required  to  divide  — — r-  by  - — ; — . 

^  x-\-b    ^  5a?+a 

__        xArCi     hx-\-a     ^x"-{-^ax-\-d?'    . 

4.  It  is  required  to  divide  -r-; — :,  by  -^, — . 
H       — ^       a:4-ct__2a;^+a)  __         2x 

5.  It  is  required  to  divide  —  by  -.  Ans. — ^, 

5      -^  X  15 

4x^  4a; 

6.  It  is  required  to  divide  -~  by  5>r.  Ans. — . 

/  oo 

7.  It  is  required  to  divide by  —  .  Ans.'      ■  . 

6  3  4x 

3.  It  is  required  to  divide by  -,  Ans -. 

1 — X    '5  1  — r 

^(IX'T~X^  X 

0.  It  is  required  to  divide  ^— — -~  by . 

c^  —  x^      -^  c—x 


Ans. 


c^+cx-{-x^' 


10.    It   is    required    to   divide -— —  by— -^, 

x-'^2bx-tb-     -^   x—b 


A 


ns. 


Also,  when  the  two  numerators  or  the  two  denominators,  can  be  divided 
by  some  common  quantity,  that  quantity  may  be  thrown  out  of  ea^-h,  gnfljtfte 
quotirnts  used  instead  of  the  fractions  first  proposed. 


u 


INVOLUTION 


INVOLUTION. 

Involution  is  the  raising  of  powers  from  any  proposed 
root ;  or  the  method  of  finding  the  square,  cube,  biquad« 
rate,  &c.  of  any  given  quantity. 

RULE  I. 

Multiply  the  index  of  the  quantity  by  the  index  of  th£ 
power  to  which  it  is  to  be  raised,  and  the  result  will  be  the 
power  required. 

Or  multiply  the  quantity  into  itself  as  many  times  less 
one  as  is  denoted  by  the  index  of  the  power,  and  the  last 
product  will  be  the  answer. 

J^ote.  When  the  sign  of  the  root  is  -f->  all  the  powers 
of  it  will  be  +  ;  and  when  the  sign  is  — ,  all  the  even 
powers  will  be  4-,  and  the  odd  powers  — :  as  is  evident 
from  multiplication.* 

EXAMPLES. 


a,  the  root. 
a^=r  square. 
a^=cube. 
a'*=4th  power. 
a^=5th  power. 
&c. 


— 3a  the  root. 
-}-9a^=square. 
— 27a^=cube. 
-f  81a'*=4th  power. 
&c. 


a^  the  root. 
a'*=  square. 
a^==cMbe. 
a^=4th  power. 
a>  «=5th  power. 
&c. 


—  2ax^  the  root. 
-|-4aV= square. 
— 8a^z^=cube. 
-f- 1 6a  V= 4  th  power. 
&c. 


*  Any  power  of  the  product  of  two  or  more  quantities  is  equal  to  tht 
same  power  of  each  of  the  factors  multiplied  together.  And  any  power 
of  a  fraction  is  equal  to  the  same  power  of  the  numerator  divided  by  the 
like  power  of  the  denominator. 

Also,  a^  raised  to  the  nth  power  is  a^'^  ;  and  —a^  raised  to  the  nt!. 
twwer  is  -J^  n"^",  according  as  n  is  an  even  or  an  odd  numlber. 


mVOLUTlON. 


45 


-  the  root. 
a 

-j=squarc. 
-r=cube. 


^=4th  power. 


&c. 


X — a  the  root. 
X — a 

x^ — ax 

y)^—2ax-Ta^  square, 
r  —  a 

x^ — 'i:ax^-\-a^x 
— ax^^-^-io^x  —  a^ 

r^— 3ax^-t-3o^x— a'  cube. 


_2^  the  root. 

4a^x* 


8a3a:« 


--—cube. 


2763 

16a4a:3 

-^^-=4th  power. 

&c. 


x+a  the  root, 
x+a 


x^H-2ox+a^  square. 

X+tt 


'-j-riax^-t-^a-^x+a^  cube. 


EXAMPLliS  FOR  PRACTICE. 

1 .  Required  the  cube  or  third  power,  of  ?a^ 

Ans.  Sa'. 

2.  Required  the  biquadrate,  or  4th  power,  of  Sa^x. 

Ans.  16aV. 

2 

3.  Required  the  cube,  or  third  power,  of— -x^. 


Ans 


-J ,  Required  the  biquadrate,  or  4th  power,  of 


27 

3a2x 
562  * 


xy. 


Ans. 


81aV 
625F 


i6  INVOLUTION. 

5.  Required  the  4th  power  of  o-f-x  ;  and  the  5th  pow- 
er of  a  -  y.         Ans.  a'*-f***fl^a:-f-6aV-}-4aa;^4-*''i  and  a'— 
6o'^2/+  lOaY  -  lOoYH-Saj/*— 3/^ 

RULE  II. 

A  binomial  or  residual  quantity  may  also  be  readily  rais- 
ed to  any  power  whatever,  a-*  follows  : 

1 .  Find  the  terms  without  the  coefficients,  by  observing 
that  the  index  of  the  first,  or  leading  quantity,  begins  with 
that  of  the  given  power,  and  decreases  contmually  by  I^ 
in  every  term  to  the  last ;  and  that  ia  the  following  quan- 
tity, the  indices  of  the  terms  are  1,  2,  3,  4,  &c. 

2.  To  find  the  coefficients,  observe  that  those  of  the 
first  and  last  terms  are  always  1  ;  and  that  the  coefficient 
of  the  second  term  is  the  index  of  the  power  of  the  first : 
and  for  the  rest,  if  the  coefficient  of  any  term  be  multi- 
plied by  the  index  of  tho  leadmg  quantity  in  it,  and  the 
product  be  divided  bv  the  .-mmber  of  terms  to  that  place, 
it  will  give  tf.e  coefficient  of  the  term  next  following. 

Note,  The  whole  number  of  terms  will  be  one  more 
than  the  index  of  the  given  power  ;  and  when  both  terms 
of  the  root  are  -f-,  all  the  terms  of  the  power  will  be  -|-  ; 
but  if  the  second  term  be  -,  all  the  odd  terms  will  be  -f , 
and  the  even  terms  ~  ;  or,  which  is  the  same  thing,  the 
terms  will  be  -f  and  -  alternately-* 


*  The  rule  here  given,  which  is  t  ;e  same  in  the  cases  of  integral  powers 
as  in  the  biaomial  theorem  of  N^jvion,  may  be  expressed  in  general  terms 
as  follows  ; 
(a+6)    «=o    ^.tnar  \h-rrn—-a     ab2  ■hm—-.  —^a     363,  &,c. 

(a-6)"»  =r  a^-ma'»-i6  ^m.^a"»a62_m.!!i=i.-7^a*"-363,  &c. 

2  2         3 

which  formulae  will  also  equally  hold  when  m  is  a  fraction,  as  will  be  more 
fully  explained  hereafter 

It  may,  also,  be  farther  obser\"ed,  (hat  the  sum  of  the  coefficients  in  every 
power,  is  equal  to  the  number  'i  raised  to  that  power.  Thus  1  +  1=2,  for 
the  first  power;  1  4-24- 1  =4=22,  for  the  square  j  I -f-5 -j-S-f-l  «=8 
s5  g3,  for  the  cube,  or  third  power  ;  and  so  on. 


EVOLUTION.  4^ 


EXAMPLES. 

1.  Let  a+x  be  involvedt  or  raised  to  the  5th  power. 
Here  the  terms,  without  the  coefficients,  are 

a^  a^ar,  o^x^,  aV,  ax\  x^. 
And  the  coefficients,  according  to  the  rule,  will  be 
5X4    10X3    10X2   6X1 

or  1,  5,  10,         10,  5  1, 

Whence  the  entire  5th  power  of  a-^x  is 

a5+5a^x+  I0aV+  lOaV+Sax^+x^. 
3.  Let  a — ,r  be  involved,  or  raised,  to  the  6th  powen 
Here  the  terms  without  their  coefficients  are 

a^  a^Xj  •i'*a:^,  aV',  a^r\  ax\  x^. 
And  the  coefficients,  found  as  before,  are, 

6X5    15X4    20X3    15X2   6X1 
'  ^'  ~2'~'  ~3~'  ~T~'  T~^     6     ' 
or  1,6,    15,        20,         16,  6,  1. 

Whence  the  entire  6th  power  of  a-^r  is 

a«-  6a^a:-f  15aV  -  20a  .r'-j-l-^a'jt*     6flar5_f-a;^ 

3.  Required  the  4th  power  of  aA-x,  and  the  .^th  power 
o^  a  —  x,  A.is.  a  +4a  a;4-6a  x^4-4'»xH-a^S  anda^  — 

4.  Required  the  6th  power  of  a  f  6,  and  th<'  7th   power 
of  a  -1/.  Ans.  <i"  4  6a '6  f   5a  6''-i-20tt  6^+  1  ^a^i'-l- 

6a6  -^^^  and   »    -  7a'</-f  2  la^y^— 35 

5.  Required   the  5th    power  of     -{-a:,  and    the  cube  of 
•3  — Aa:-|-c.  Ans.    ■><J  +  .SUx+80a; -|-4()x'-M0r''-f  a:^  and 

a^  -f  .ia^c 4-  <af  +  c'  —  .■3a^6x -  &achx  — 
3c"^6x  -f3a6  V-(-  :3c6  V-  6  V. 


EVOLUTION. 

Evolution,  or  the  extraction  of  roots,  is  the  reverse  of 
involution,  or  the  raising  powers  ;  being  the  method  of  find- 
ing the  square  root,  cube  root,  &c.  of  any  given  quantity. 


4S  EVOLUTION. 

CASE  I. 
To  find  any  root  of  a  simple  quantity. 

RULE. 

Extract  the  root  of  the  coefficient  for  the  numeral  part^ 
and  the  root  of  the  quantity  subjoined  to  it  for  the  literal 
part ;  then  these,  joined  together,  will  be  the  root  re- 
quired. 

And  if  the  quantity  proposed  be  a  fraction,  its  root  will 
be  found  by  taking  the  root  both  of  its  numerator  and  de- 
nominator. 

Kote.  The  square  root,  the  fourth  root,  or  any  other 
even  root,  of  an  affirmative  quantity,  may  be  either  4"  or 
— .  Thus  v^a"=-f  a  or  —a,  and  {/6*  =  -}-6  or  --6,  &c. 
But  the  cube  root,  or  any  other  odd  root,  of  a  quantity, 
will  have  the  same  sign  as  the  quantity  itself.     Thus, 

\/a^=a\   \/—a^— — a  ,  and  %/^a^=-^a^  &c.* 

It  may  here,  also,  be  farther  remarked,  that  any  even 
root  of  a  negative  quantity  is  unassignable. 

Thus,  ^  —  a^  cannot  be  determined,  as  there  is  no 
quantity,  either  positive  or  negative,  (+  or  — ),  that  when 
multiplied  by  itself,  will  produce  — a^. 

EXAMPLES. 

i.  Find  the  square  root  of  9a;- ;  and  the  cube  root  of 

Here  x/^x^=\/^  X ^x''=SXx=3x.     Ans. 
And    ^8a;^=^8X^a;^=2Xx=2a\     Ans. 


*  The  reason  why  -f  a  and  — a  are  each  the  square  root  of  a2  is  obvious, 
since  by  the  rule  of  multiplication,  (_|-o)X("f-«)  and  ( — a)X(— o)  are 
both  equal  to  a2. 

And  for  the  cube  root,  fifth  root,  &c.  of  a  negative  quantity,  it  is  plain, 
from  the  rame  rule,  that 
i— a)X(— a)X(— o)  =  — a3  ?  and  (— a3)  x(-ho2)= —aS, 

And  consequently  y/—n3  =  —a,  and  y/~n5'=—a. 


EVOLUTION.  4^ 


Orsr 
t.  It  is  required  to  find  the  square  root  of-~— 


'ibe  cube  root  of— 


4c^ 


21  c" 

3.  It  is  required  to  tind  the  square  root  of  4aV. 

Ans.  2aa;^ 

4.  It  is  required  to  find  the  cube  root  of-  125aV. 

Ans.  ~5aic^. 

5.  It  is  required  to  find  the  4th  root  of  266a*a:^ 

Ans.  4aa;^o 

6.  It  is  required  to  find  the  square  root  of - 


Ans. 
7.  It  is  required  to  find  the  cube  root  of-— -g-. 


8,  It  is  required  to  find  the  5th  root  of  • 


Ans. 
32flV«_ 
243~' 

Ans. 


^1 

2a 

'2ax^ 


3 

CASE  IL 
To  extract  the  square  root  of  a  compound  quantity^ 

RULE. 

\,  Range  the  terms,  of  which  the  quantity  is  composed^ 
according  to  the  dimensions  of  some  letter  in  thera,  begin- 
ning with  the  highest,  and  set  the  root  of  the  first  term  ia 
the  quotient. 

2.  Subtract  the  square  of  the  root  thus  found,  from  the 
first  term,  and  bring  down  the  two  next  terms  to  the  re- 
mainder for  a  dividend. 


aO  EVOLUTION. 

3.  Divide  the  dividend,  thus  found,  by  double  that  part 
of  the  root  already  determined,  and  set  the  result  both  in 
the  quotient  and  divisor. 

4.  Multiply  the  divisor,  so  increased,  by  the  term  of 
the  root  last  placed  in  the  quotient,  and  subtract  the  pro- 
duct from  the  dividend ;  and  so  on,  as  in  common  arith- 
metic. 

EXAMPLES. 

1.  Extract  the  square  root  of  rr'*  — 4a;^-f6x2— 4a;-f-l 
a:* -4r^-f6«'--4af +I(x2  -  2x4-1 


2a;2— 2a;)-4a='-}-6a;^ 
-4a;3-|-4a:2 

2x2— 4a;-|-i)2a;2-4a;-}-i 
2x2— 4x-fl 


Ans.  x^ — 2x-{-l,  the  root  required. 
2.  Extract  the  square  root  of  4a*+12a='x-{-13aV-f  6 

TX^-f-X^ 

.    4fl^-f  iSa^^x^- l3ttV-l-6ax='-l-x^(2a=+3ax'i-x^ 
4a^ 


4a2-f3ax(  12a='x+ 1 3a  V 
12a='x-l-9aV 


4a2+6ax-|-j-)4a=x2-l-6ax3+x^ 
4aV-f6ax'+x^ 


.\ote.  When  the  quantity  to  be  extracted  has  no  exact 
coot,  the  operation  may  be  carried  on  as  far  as  is  thought 
accessary,  or  till  the  regularity  of  the  terms  shows  the  law 
by  which  the  series  would  be  continued. 

EXAMPLE. 

h  It  is  required  to  extract  the  square  root  of  1+x. 


EVOLUTION.  5t 


H-.(.+---+---^,&c. 

1 


'+!) 


a:- 


■  +  .—,)- 


4 


x2     a;3     a;* 
"'T""¥     64 


^  I         X*      x' 

24-x 

^         4       16/  8      64 


)x^ 
"8" 


Y'*"T6""64"^256 

"T     64  ""256 

Here,  if  the  numerators  and  denominators  of  the  tiyo 
fast  terms  be  each  multipHed  by  3,  which  will  not  alte? 
their  values,  the  root  will  become 

X     a:g        'iix2         3.5x4  3.5.73-5 

^"^2     2.4     2.4.6      2.4.6.8'^2.4.6.8.10* 
^vhere  the  law  of  the  series  is  manifest. 

EXAMPLES  FOR  PRACTICE. 

2.  It  is  required  to  find  the  square  root  of  a*-|-4a^a;+ 
6oV-f4ax3-f a:^  Ans.  a»+2ax+«*, 

3.  It  is   required  to  find   the  square  root  of  x*  — 2a;=*-f 

g  1 

^x2-ix+— .  Ans.  a;»— x4-J. 


^  EVOLUTION. 

4.  It  is  required  to  find  the  square  root  of  4a;^— 4xH 
32r'-f-x2-  6a;+9.  Ans.  2x'^x-}-3. 

5.  Required  the  square  root  of  x^-f-4x^+«Ox*-}-20x^-f 
26x2+24x4.16.  Ads.  x='-h2r^+3x+4 

6»  It  is  required  to  extract  the  square  root  of  a^-f^. 

Ans.  a-i- -r-l — tr,  &c. 

7.  It  is  required  to  extract  the  square  root  of  2,  or  of 
^  +  1'  Ans.  i+^-i+_V-3V-i-^V,  &G 

CASE  III. 

To  find  any  root  of  a  compoimd  quantiiy. 

RULE. 

Find  the  root  of  the  first  term,  which  place  in  the  quo- 
tient ;  and  having  subtracted  its  corresponding  power  from 
that  term,  bring  down  the  second  term  for  a  dividend. 

Divide  this  by  twice  the  part  of  the  root  above  deter- 
mined, for  the  square  root ;  by  three  times  the  square  of 
it,  for  the  cube  root,  and  so  on  ;  and  the  quotient  will  be 
the  next  term  of  the  root. 

Involve  the  whole  of  the  root,  thus  found,  to  its  proper 
power,  which  subtract  from  the  given  quantity,  and  divide 
the  first  term  of  the  remainder  by  the  same  divisor  as  be- 
fore ;  and  proceed  in  this  manner  till  the  whole  is  finish- 
ed.* 


*  As  this  rule,  in  high  powers,  is  often  found  to  be  very  laborious,  it  mav 
be  proper  to  observe,  that  the  roots  of  various  compound  quantities  niaj 
sometimes  be  easily  discovered,  as  follows  : 

Extract  the  roots  of  all  the  simple  terms,  and  connect  them  together  b\ 
the  signs  -}-  or  — ,  as  may  be  judged  most  suitable  for  the  purpose  ;  thei. 
involve  the  compound  root,  thus  found,  to  its  proper  power,  and  if  it  be  the 
same  with  the  given  quantity,  if  is  the  root  required.  But  if  it  be  found  tc 
differ  only  in  some  of  \he  signs,  change  them  from  -f-  to  — ,  or  from  — -  to 
-f-,  till  its  power  agrees  with  the  given  one  throughout. 

Thus,  in  the  third  example  next  foilowing,  the  root  is  2a — 3x,  which  is 
the  difference  of  the  roots  of  the  first,  and  last  terms:  and  in  the  fourth  ex- 
ample, the  root  is  a-J-6-}-c,  which  is  the  sum  of  the  roots  of  the  first,  fourth;, 
and  sixth  terms.  The  same  may  also  be  observed  of  the  sixth  example 
where  the  root  is  found  frera  the  first  and  last  terms. 


EVOLUTION.  5S 


EXAMPLES. 


1.  Required  the  square  root  of  o* — 2a^a;4-3aV'-2ar 
a' 


2a')— 2ti^x 


2ay2a'x' 


* 

2.  Required  the   cube   root  of  a'*-|-6a;^—  40r'4*96a:'— 

a:6-  6x5— 40x34-96x-64(.x2-(-2x-4 

Sx^ex^ 

x«+6x5+12x^+8x^ 
3x*)-"  12x« 

a;6-|_6x5— 40xS^96x— -64 

* 

3.  Required  the  square  root  of  4a^— 12ax-f-9:»''^. 

Ans.  2a-=-3x, 

4.  Required  the  square  root  of  a-4"2a6-i-2ac+62-4-26c< 
■fc^  Ans.  a+6-{-c. 

5.  Required  the   cube  root  of  x^— 6x^4-1 6x^— 20x^4- 
JSx-^—ex-f-l.  Ans.  X-— 2x+l. 

6.  Required   the  4th  root  of  16a*-.96a='x+216aV~ 
216«x^+81xS  An5.2a-3i^. 

F  2 


■A    IRRATIONAL  QUANTITIES,  on  SURDS. 

7.  Required  the   6th  root   of  32J^;5~80x^-{-80a:^-40«• 
^10a;-I.  Ans.  2x-1 

Of  irrational  QUANTITIES, 
OR  SURDS. 


Irrational  Quantities  or  Surds,  are  those  of  which 
ihe  values  cannot  be  accurately  expressed  in  numbers ; 
^nd  are  usually  expressed  by  means  of  the  radical  sign  ^, 
or  by  fractional  indices  ;  in  which  latter  case,  the  nume- 
rator shows  the  power  the  quantity  is  to  be  raised  to,  ancr 
the  denominator  its  root. 

Thus,  \/2f  or  2^,  denotes  the   square  root  of  2  ;  ^«r 

ora^f  the  cube  root  of  the  square  of  g,  &c.* 

CASE  I. 

To  reduce  a  rational  quantity  to  the  form  of  a  iurd. 

RULE. 

Raise  the  quantity  to  a  power  corresponding  with  tha. 
denoted  by  the  index  of  the  surd  ;  and  over  this  new 
quantity  place  the  radical  sign,  or  proper  index,  and  it  wih 
he  of  the  foxm  required. 

EXAMPLES. 

!,  Let  3  be  reduced  to  the  form  of  the  square  root. 
Here  3X3=3^=9  ;  whence  ^/9.     Ans. 


*  A  quantity  of  the  kind  here  mentioned,  as  for  instance  *y  2,  is  called  a:, 
.rralional  number,  or  a  surd,  because  no  number,  either  whole  or  fractional 
can  be  found,  which,  when  multiplied  bj  itself,  vpill  produce  2.  But  its  ap 
proximate  value  may  be  determined  to  any  degree  of  exactness,  by  the  coni- 
men  rule  for  extracting  the  square  root,  being  1  and  certain  non-per'odic  <^e 
■^^mals,  which  never  terminate 


IKRATIONAL  QUANTITIES,  or  SURDS.     5^ 

2.  Reduce  2a;^  to  the  form  of  the  cube  root. 

Here  {2xy=6x^ ;  whence  ^/Sx^  or  (8a:«)^. 

3.  Let  5  be  reduced  to  the  form  of  the  square  root. 

Ans.  ^/(25) 

4.  Let  — 3a;  be  reduced  to  the  form  of  the  cube  root. 

Ans.  y-(27a:-''), 

5.  Let  —2a  be  reduced  to  the  form  of  the  fourth  root. 

Ans.  ■—{/  (I6a*). 

6.  Let  a^  be  reduced  to  the  form  of  the  fifth  root,  and 

x/a-^-^b,  ^-—  and  ; to  the  form  of  the  square  root. 

^    '   2a  b^a  ^  „ 

Ans.  {/a^",  v/(a+2v^a6-|-6),  ^/{}a),  and  ^6'^. 

JVote.    Any  rational  quantity  may  be  reduced  by  the 

above   rule,  to  the  form  of  the   surd  to  which  it  is  joined, 

and  their  product  be  then  placed   under  the  same  index.. 

or  radical  sign. 

EXAMPLES. 

Thus  2^/2  =  ^4 Xv/2  =  ^4X2=v^8 
And   23/4  =  y8Xy4=y8X4  =  y32 
Also  3^a=y^9Xv/a=v/9Xa=v^9a 
And  |,V4a=^i=^4a=3/i.  x  4a= Vf 

1.  Let  Sy/G  be  reduced  to  a  simple  radical  form. 

Ans.  v/(lfiQ}^ 

2.  Let  iy^Sa  be  reduced  to  a  simple  radical  form. 

Ans.  v/(f). 
-      2a       9 

3.  Let  -^Vtt  ^®  reduced  to  a  simple  radical  form. 

Ans.  y— , 

CASE.  IL 

To  reduce  quantities  of  different  indices,  to  others  that 
shall  have  a  given  index. 

RULE, 

Divide  the  indices  of  the  proposed  quantities  by  th«^ 


56    IRRATIONAL  QUANTITIES  or  SURDS. 

given  index,  and  the   quotients  will  be  the  new  indices  for 
those  quantities. 

Then,  over  the  said  quantities,  with  their  new  indices ^ 
place  the  given  index,  and  they  will  be  the  equivalent 
quantities  required. 

EXAMPLES. 

1 .  Reduce  3^  and  2  ^  to  quantities  that  shall  have  the 
"  index  i. 

1  1        1        fi       R 

Here -~-=-X-=-=3,  the  1st  index: 

2  6     2     12 

AVhence  (3^)«  and  (23)e,  or  27^,  and  4«,  are  the  quae 
tities  required. 

2.  Reduce  6^  and  6^  to  quantities  that  shall  have  the 

1  1  1 

common  index  -.  Ans.  1256  and  36  «. 

6 

3.  Reduce  22  and  4*  to  quantities  that  shall  have  the 

1  1  i 

common  index  -.  Ans.  16^  and  16»' 

8 

4.  Reduce  a^  and  a^  to  quantities   that  shall  have   the 

1  L  2 

common  index  -.  Ans.  {a^y  and  (a^)*. 

5.  Reduce  a^  and  b^  to  quantities  that  shall  have  the 
common  index  -.  Ans.  (a^)*  and  (6*)« 

JVote.  Surds  may  also  be  brought  to  a  common  index, 
by  reducing  the  indices  of  the  quantities  to  a  common  de- 
nominator, and  then  involving  each  of  them  to  the  power 
denoted  by  its  numerator. 


IRRATIONAL  QUANTITIES,  or  SURDS.    5T 

EXAMPLES.* 

1.  Reduce  3i  and  4^  to  quantities  having  a  common  in 
dex. 

Here  32=3^=(33)^=(27)^ 

And  4^=4«=(4  2)«=(16)^ 

Whence  (27)6  and  (i6)6.  Ans. 

2.  Reduce  4^  and  5*  to  quantities  that  shall  have  a  com 
mon  index. 

Ans.  256^'^  and  Ub^^. 

3.  Reduce  a^  and  o^  to  quantities  that  shall  have  a  com= 
mon  index. 

Ans.  (a3)«and(aa)». 

4.  Reduce  a 3  and  6*  to  quantities  that  shall  have  a  com 
mon  index. 

Ans.  (a*)'*'^and(63)TV 

6.  Reduce  a»  and  b"*  to  quantities  that  shall  have  a  com- 
mon index, 

Ans.  (o"»)^  and  (6")^* 

CASE  III. 

To  reduce  surds  to  their  most  simple  forms, 

RULE. 

Resolve  the  given  number,  or  quantity,  into  two  factors^ 
one  of  which  shall  be  the  greatest  power  contained  in  it, 
and  set  the  root  of  this  power  before  the  remaining  part, 
with  the  proper  radical  sign  between  them.* 


*  When  the  given  surd  contains  no  factor  that  is  an  exact  power  of  the 
kind  required,  it  is  already  in  its  most  simple  form. 
Thus,  \/15  cannot  be  reduced  lower,  because  neither  of  its  factors,  5,  nor 


58    IRRATIONAL  QUANTITIES,  or  SURDS. 


EXAMPLES. 

1.  Lei  ^48  be  reduced  to  its  most  simple  form. 

Here  ^^8=y'r6~><  3=4^3  Ans. 

2.  Let  ^108  be  reduced  to  its  most  simple  form. 

Here  »/ i Ob =(/27>< 4=3^4  Ans. 
J^ote  1.  When  any  number,   or  quantity,  is  prefixed  it 
the  surd,  that  quantity  must  be  multiplied   by  the  root  ol 
the  factor  abovementio^ied,  and  the  product  be  then  joined 
to  the  other  part,  as  before. 

EXAMPLES. 

K  Let  2  v/ 32  be  reduced  to  its  most  simple  form. 
Here  2^-32=1^^16X2=8^2  Ans. 

2.  Let  6^24  be  rfd'jced  to  its  most  simple  form. 
Here  5^/  '4=53/8X3=  10^3  Ana. 

JVote  2.  A  fractional  surd  may  also  be  reduced  to  a  more 
convenient  form,  by  multiplying  both  the  numerator  and 
denominator  by  such  a  number,  or  quantity,  as  will  make 
the  denominator  a  complete  power  ol  the  kind  required  ; 
and  then  joining  its  root,  with  1  put  over  it,  as  a  numera. 
tor,  to  the  other  part  of  the  surd.* 

EXAMPLES. 

2 
1 .  Let  v^-  be  reduced  to  its  most  simple  form. 


*  The  utility  of  reducing  surds  to  their  most  simple  forms,  in  order  to  have 
(he  answer  in  decimals,  will  be  readily  perceived  from  considering  the  first 
question  above  given,  where  it  is  found  that  ^^  =3  ^y/ 14  ;  in  which  case 
it  is  only  necessary  to  extract  the  square  root  of  the  whole  number  14,  (or  to 
find  it  in  some  of  the  tables  that  hare  Been  calculated  for  this  purpose)  and 
then  divide  it  by  7  ;  whereas,  otherwise,  we  must  have  first  divided  the  nu- 
merator by  the  deaomioa^or,  and  then  have  found  the  root  of  the  quotient,  for 
the  surd  part ;  or  else  have  determined  the  root  both  of  the  numerator  and 
denominator,  and  then  divided  one  by  the  other  ;  which  are  each  of  them 
troublesome  processes  when  performed  by  the  common  rules ;  and  in  the  nex* 
eaample  for  the  cube  root,  the  labour  would  be  much  greater. 


IKRATIONAL  QUANTITIES,  on  SURDS.    59 

Here  ^l=y±=sj  (Ix  u)  =1^14  Ans. 

2 

1,  Let  3^-  be  reduced  to  its  most  simple  form. 

aere3i/|=3y/l=3v(-Lx60)=?y50.  Ans, 

EXAMPLES  FOR  PRACTICE. 

3.  Let  V  J  25  be  reduced  to  its  most  simple  form. 

Ans.  5v^5< 

4.  Let  v^294  be  reduced  to  its  most  simple  form. 

Ans.  7^6, 

5.  Let  ^56  be  reduced  to  its  most  simple  form. 

Ans.  23/7. 

6.  Let  ^192  be  reduced  to  its  most  simple  form. 

Ans.  4^3. 

7.  Let  7v/  80  be  reduced  to  its  most  simple  form. 

Ans.  28  yj  5. 

8.  Let  9^81  be  reduced  to  its  most  simple  form. 

Ans.  273/3. 

3  5 

9.  Let  --^rzx/z:  be  reduced  to  its  most  simple  form. 

121^  6  ^ 

Ans.  slav/SOi 

4  3 

i  0.  Let  ~\/t^  be  reduced  to  its  most  simple  form. 

Ans.  4yT2. 

11.  Let  ^98a^a;  be  reduced  to  its  most  simple  form. 

Ans.  7ay/2i. 

12.  Let  ^x^-'a^x^  be  reduced  to  its  most  simple  form. 

Ans.  a;v'(a;  — a^}. 

CASE  IV. 

To  add  surd  quantities  together. 

RULE. 

When  the  surds  are  of  the  same  kind,  reduce  them  to 


^0    IRRATIONAL  QUANTITIES,  or  SURDS. 

their  simplest  forms  as  in  the  last  case  ;  then,  if  the  surd 
part  be  the  same  in  them  all,  annex  it  to  the  sum  of  the 
rational  parts,  and  it  will  give  the  whole  sum  required. 

But  if  the  quantities  have  different  indices,  or  the  surd 
part  be  not  the  same  in  each  of  them,  they  can  only  be 
added  together  by  the  signs  -\-  and  — . 

EXAMPLES. 

U  It  is  required  to  find  the  sum  of  ^27  and  ^48. 
Here  v/27=v/ftX3=  3^3 
And   -v/48=-v/ 1 6X3=4^/3 

Whence  T  v  3  the  sum. 
2^  It  is  required  to  find  the  sum  of  ^600  and  ^  108, 
Here  ^500=^ 3/ 7^^X4=53/4 
And    ^10S=y~27"X4=3V'4 

Whence  S\/4  the  sum. 

3.  It  is  required  to  find  the  sum  of  4^147  and  3 
,/76.  

Here  4^' 147  =  4^49^X3=28^3 
And    3^  75=3^25'x3=i5v'3 

Whence  43^3  the  sum. 

2  1    - 

4.  It  is  required  to  find  the  sum  of  Sy'-  and  2-^/-—, 

2  10     3 

Here  3v'-=3^--=-^10 

And   vi=2^iS5=-^^/">• 


4 
Whence  -y/  i  0  the  sum. 


IRRATIONAL  QUANTITIES,  or  SURDS.    61 

EXAMPLES  FOR  PRACTICE. 

5.  It  is  required  to  find  the  sum  of  y72  and  y'128, 

Ans.  14-^(2), 

6.  It  is  required  to  find  the  sum  of  y' 180  and  v^405. 

Ans.  «5v^(5). 

7.  It  is  required  to  find  the  sum  of  3^40  and  ^135. 

Ans.  9^(5), 
S.  It  is  required  to  find  the  sum  of  4^54  and  5^128. 

Ans.  32^  (2), 

9.  It  is  required  to  find  the  sum  of  9^243  and  10  ^^  363, 

Ans.  191^(3), 
2  27 

10.  It  is  required  to  find  the  sum  of  Sy'-  and  '^y'-— 

Ans.  3tVy(6)^ 

11.  It  is  required  to  find  the  sum  of  12^-  and  3^— 

Ans.  6f  V(2)'. 

12.  It   is  required   to  fii  d  the  sum  of  a  ^/a^b  and 

Av'46x*.  Ans.  (^+^yV'*" 

CASE  V. 

To  find  the  difference  of  surd  quantities, 

RULE, 

When  the  surds  are  of  the  same  kind,  prepare  the  qua£^ 
titles  as  in  the  last  rule  ;  then  the  difference  of  the  rational 
parts  annexed  to  the  common  surd,  will  give  the  whole  dif* 
Terence  required. 

But  if  the  quantities  have  different  indices,  or  the  surdi 
part  be  not  the  same  in  each  of  them,  they  can  only  b9 
subtracted  by  means  of  the  sijrn  — . 

1.  It  is  required  to  find  the  difference  of  ^^448  aoi 

./n2. 


62    IRRATIONAL  QUANTITIES,  or  SURDfe 


Here  y44S=^v^64X 7=8^/7 
And    ^1 12=^76X7=4 v/7 

Whence  4v^7  the  difference. 

2.  It  is  required  to  find  the  difference  of  ^192  and 
i/24.  

Here  y  192  =  V^4>< 3=43/3 
And   ^24  =3/~bx"oi=2V3 

AVhence  2^3  the  difference. 

3.  It  is  required  to  find  the  difference  of  5v'20  and 
3v/45,  

Here  5^20=5^4  X  5=10y5 
And    3-s/45=3^9X5=  9v/5~ 

Whence  v^6  the  difference. 

3      2 

4.  It  is  required  to  find   the  difference  of--/-, ana 


5^6' 


„       3      2      3      6        3      .        I    .. 
«^^^-4^3  =  4^9  =  r2^'^4^' 


lerence,  or  answer  required. 


Whence  ~^/6  the  dit 


EXAMPLES  FOR  PRACTICE. 

1.  It  is  required  to  find  the  difference  of  2y^o0  ana 


IRRATIONAL  QUANTITIES,  or  SURDS.    6S 

2.  It  is  required  to  find  the  difterence  of  ^320  and 
»^/4().  Ans.  23/(5). 

3  6 

3.  It  is  required  to  find  the  diiference  of  v^rand^-r. 

Ans. -,%^{\b), 

4.  It  is  required  to  find  the  difference  of  2^|  and  ^8> 

Ans.  y/{2). 

5.  It  is  required  to  find  the  difference  of  3^^  and  ^72, 

Ans.  y{9). 

2  9 

6.  It  is  required  to  find  the  difference  of  I/-  and^^^— s 

3  82 

Ans.  J-3y(l«). 

7.  It  is  required  to  find  the  difference  of  ^Soa^x  and 
V20aV.  Ans.  Ua^—2ax)y(bx), 

8.  It  is  required  to  find  the  differ^^nce  of  8^a^6  and 
2ya^6.  Ans.  (8a— 2a') y (6). 

JVoie,  The  two  hist  answers  may  b*-  written  thus, 
(2aa; — 4a^)y^5a;),  and 
(2a2-8a)y(6), 

CASE  VI. 

2o  multiply  surd  quantities  together, 

RULE. 

When  the  surds  are  of  the  same  kind,  find  the  product 
of  the  rational  parts,  and  the  product  of  the  surds,  and 
the  two  joined  together,  with  their  common  radical  sign 
between  them,  will  give  the  whole  product  required; 
which  may  be  reduced  to  its  most  simple  form  by  Case  in. 

But  if  the  surd.s  are  of  different  kinds,  they  must  be 
reduced  to  a  common  index,  and  then  multiplied  together 
as  usual. 

It  is  also  to  be  observed,  as  before  mentioned,  that  the 
product  of  different  powers,  or  roots  of  the  same  quantity, 
is  found  by  adding  their  indices. 


o4    IRRATIONAL  QUANTITIES,  or  SURDS. 


EXAMPLES. 

1.  It  is  required  to  find  the  product  of  3-^/8  and  2^b 

Here  3y8 

Multiplied  2^6 

Gives        t)v^48=6^  16X3=24-/ 3    Ans. 

12        3      6 

2,  It  is  required  to  find  the  product  of  -  Vq  ^"^7  >/«- 

Here         I^| 
Multiplied -y  I 


_.  3      10      3      6      3      15 

3.  It  is  required  to  find  the  products  of  2^  and  3^. 
Here  22=26  ==(2  )^  =  8^ 


And    3^=3e=(32)^=9» 


Whence  (72)  J-  Ans. 
4  It  is  required  to  find  the  product  of  5^a  and  3^fl 

Here  6./a:=5o-=5a6 

And    33/a=3a3=3a6 


Whence  15ae  =  15  {a^y  or  ISya^  Ans- 


EXAMPLES  FOR  PRACTICE. 


6.  It  is  required  to  find  the  product  of  5^8  and  3^/5. 

Ans.  30^(10), 
o.  It  is  required  to  find  the  product  of  V^S  and  5y4. 

Ans>  10y\^9), 


IRRATIONAL  QUANTITIES,  or  SURDS.    6S 

I  2 

7.  Required  the  product  of  --^6  and  —z^/^. 

Ans.  xV-v/CS). 
9.  Required  the  product  of --/IS  and  5^2:0. 

Ans.  15^/(10). 

9.  Required  the  product  of  2^3  and  13|-v/5. 

Ans  27^(15). 

10.  Required  the  product  of  T2\a^  and  VlO^^a*.       ^^ 

Ans."8706ia»2^ 

11.  Required  the  product  of  4+2^/2  and  t—^2. 

Ans.  4. 

12.  Required  the  product  of  (a -ft)''  and  (a-f6)"'. 

Ans.  {a-\-h)lKTr 

CASE  VII. 
To  divide  one  surd  quantity  by  another, 

RULE. 

\Vhen  the  surds  are  of  the  same  kind,  find  the  quotient 
jI  the  rational  parts,  and  the  quotient  of  the  surds,  and  the 
iwo  joined  together,  with  their  common  radical  sign  be- 
t'.veen  them,  will  give  the  whole  quotient  required. 

But  if  the  surds  are  of  different  kinds,  they  must  be 
reduced  to  a  common  index,  and  then  be  divided  as  before* 

It  is  also  to  be  observed,  that  the  quotient  of  different 
powers  or  roots  of  the  same  quantity,  is  found  by  subtract- 
ing their  indices. 

EXAMPLES. 

,  It  is  required  to  divide  8^/108  by  2*/6. 

8^/108  

Here~2;^=4v^  18=4^9  X  2=  12y'2  Ans, 

g2 


6^    ERRATIONAL  QUANTITIES,  on  SURDS 

2.  Itti  required  to  divide  8^612  by  4^2. 
Here  ?>0~=2^256  =2^64X4=8^4  Ans, 

S.  It  is  required  to  divide  -</5  by  ^^/2. 
It/5     3     6     3     10     3    ^.^  , 

4*  It  19  required  to  divide  v/7  by  l/7s 
g»  It  is  required  to  divide  6^64  by  3^2 


Ans.  6^b: 
Ans.  2^^^ 


S»  It  is  required  to  divide  4^/72  by  2^18. 

3        1        2     1 
f.  Itifi  required  to  divide  ^TV^r^by -^-. 

Ans.  If  v^2 
5     2         2     3 
d»  tt  ts  (required  to  divide  ^rv/^  ^X  ^--y/- . 

Ans,  ^^^2 

1  ^ 

9    Kt  U  required  to  divide  4-^fl  by  2^^c6, 

2  3 


Ans. 


2  3 

'  to.  It  i«  required  to  divide  82-^a  by  IS-^c. 

5  4 


64S  * 


3    '  9    ^ 

3i  Ic  U  is  required  to  divide  9-a    by  4— a*". 


025  »:!:« 

424 


i2.  Ct  is  required  to  divide  v/20-hv/l2  by  ^5-f-^3. 

Ans.  ^4. 
JYi^o  3taoe  the  divi^iOB  of  surds  ia  performed  by  sub 


IRRATIOISAL  QUANTITIES,  or  SURDS.    6T 

trading  their  indices,  it  is  evident  that  the  denominator  of 
any  fraction  may  be  taken  into  the  numerator,  or  the  nu- 
merator into  the  denominator,  by  changing  the  sign  of  its 
index. 

a*" 
Also,  since —  =  1,  or  =a^-'"=a°,  it  follows,  that  the 

a*" 

expression  a°  is  a  symbol  equivalent  to  unity,  and  conse 
quently,  that  it  may  always  be  replaced  by  I  whenever  it 
occurs.* 

EXAMPLES. 

]     a'^  1      a" 

J.  Thus  -=--or  a* ;  and  — =--  ,  or  o". 

b       ba-""  a"  1  6"^* 

2.  Also,  ^  =  ~,  or  6a-  ;  and  ^-:;;i=-^;i^.,  or  ^, 

3.  Let  -^  be  expressed  with  a  negative  index. 

Ans.  a-'^ 
I .  Let  o~^  be  expressed  with  a  negative  index. 

Ans.  — .. 

6.  Let — ; —  be  expressed  with  a  negative  index. 

Ans.  (a-{-x)-^ 
6.  Let  a(a^ — a;^)"^  be  expressed  with  a  negative  index. 

Ans.  — 5—-. 

*  'I'o  what  is  above  said,  we  may  also  farther  observe, 
1    That  0  added  to  or  subtracted  from  any  quantity,  makes  it  neither 
greater  nor  less;  that  is, 

a  4  0  =  a,  and  a — 0  =sa. 

2.  Also,  if  nought  be  multiplied  or  divided  by  any  quantity,  both  the  pro- 
duct and  the  quotient  will  be  nought ;  because  any  number  of  times  0,  or 
any  part  of  0,  is  0;  that  is, 

OXa.  oraxO~0,  and  «  =0. 
a 

3.  From  thi&it  likewise  follows,  that  uought  divided  by  Doughf,  is  a  finiO; 
1'iantity,  of  some  kind  or  other. 


08    IRRATIONAL  QUANTITIES,  or  SURDS 
CASE  VIII. 

To  involve^  or  raise  surd  quantities  to  any  power. 

RULE. 

When  the  surd  is  a  simple  quantity,  multiply  its  index 
by  2  for  the  square,  by  3  for  the  cube,  &c.,  and  it  will  give 
the  power  of  the  surd  part,  whicli  being  annexed  to  the 
proper  power  of  the  rational  part,  will  give  the  whole 
power  required.  And  if  it  be  a  ompound  quantity,  mul- 
tiply it  by  Itself  the  proper  number  of  times,  according  to 
the  usual  rule.* 

For  since  OXa  =0,  or  0=  OXa,  it  is  evident  that  -  =  a. 

4.  Farther,  if  any  finite  quantity  be  divided  by  0,  the  quotient  will  be 
infinite. 

For  let-«=  g,  then,  if  6  remains  the  same,  it  is  plain,  the  less  a  is,  the 

greater  will  be  the  quotient  q ;  whence,  if  a  be  indefinitely  small,  q  will  be 
indefinitely  great:  and  consequently,  when  a  is  0,  the  quotient  q  will  be  in 
iinite ;  that  is, 

Which  properties  are  of  frequent  occurrence  in  some  of  the  higher  parts  of 
:he  science,  and  should  be  carefully  remembered. 

Since,  therefore,-—  is  the  same  as  (a +6)""  .     Let  us  suppose,  in  the 

general  formula,  n*=:  —1 ;  and  we  shall  have  for  the  coefficients  n  =  — 1  ; 

n—l  n—2  n— 3 

"'2~'~ —   '  ~3  ~*^  ~^  '4 ^'  ^^'^  ^^^  for  the  powers  of  a  we 

1,         n        -1      1       n— 1         —2        1        n— 2       I       ?i— 3        1 
*5ave  a    =.a    c=-?a         c=  a       «=  —  :a         =-l,-a  -        , 

a  a2  '  a3  '  ^4  &c.  .- 

30  that  (rt +6)      ^  -_  -,-__4.__      ^      ^^^c.whichisthe 

a-^  0      a     a2       a3      a4      as     a6 
£ame  series  that  is  found  by  division.    For  more  on  this  subject  see  the  Bi- 
monial  Theorem,  (furth-.r  on)  or  Euler's  Algebra, 

When  any  quantity  that  is  aflfected  by  the  sign  of  the  square  root  is  to 
p^  raised  to  the  second  power,  or  squared,  it  is  done  by  suppressing  the  sign. 

Vo)2,or  v/aX  ^^«=«;  and  x/  (a  +  h)%  or  ^/TfT  X  7a^^  ^  » fi". 


IRRATIONAL  QUANTITIES,  or  SURDS.    69 


EXAMPLES. 

2    - 

1.  It  is  required  to  find  the  square  of -z^. 

o 

Here  (-aM2=-a=^        =:-a^=-yaK    Ans= 

2.  It  is  required  to  find  the  cube  of --v/3. 

Here  ^=3*=.J  ^  27=|.^9  X3=«y3.    Ans. 

3.  It  is  required  to  find  the  square  of  3  1/3, 

Ans.  93/9. 

4.  It  is  required  to  find  the  cibe  of  1 7-^21. 

Ans.  I03n^^{2\). 

5.  It  is  required  to  find  the  Uh  power  of -y/6. 

Ans.  ^\. 
G.  It  is  required  to  find  the  square  of  3-f  2^5. 

Ans.  rG+lSv'S. 

7.  It  is  required  to  find  the  cube  oC  ^x-{"3^y. 

Ans.  xy/a-  -f-  r,yy/z  +9xyt/+27yv^y. 

8.  It  is  required  to  find  the  4th  power  of  y^3 — ^2. 

49-^20^6, 

CASE  IX. 

To  find  the  routs  of  surd  quantities. 

RULE. 

When  the  surd  is  a  simple  quantity,  multiply  its  index 
by  \  for  the  square  root,  by  ^  for  the  cube  root,  &c.,  and 
it  will  give  the  root  of  the  surd  part ;  which  being  annexed 
to  the  root  of  the  rational  part,  will  give  the  whole  fOOi 


^0    IRRATIONAL  QUANTITIES,  or  SURDS. 

required.    And  if  it  be  a  compound  quantity,  find  its  root  b^* 
the  usual  rule.* 

EXAMPLES. 

1.  It  IS  required  to  find  the  square  root  of  9^3. 
Here  {9ydf =9^X6^^^ ^91x3^  =  ^i/\    Ans- 

2.  It  is  required  to  find  the  cube  root  of  q\/2. 

Here  (^^/''^Y  =  C-)'X{2^^^)=^(2')=^i/^.    Ans. 

3.  It  is  required  to  find  the  square  root  of  U>^. 

Ans.  iOy(I0;^ 

h 

4.  It  is  required  to  find  the  cube  root  of  ^«^- 

Ans.  |aya. 

16      2 

5.  It  is  required  to  find  the  4th  root  <>^^"^' 

Ans.  |a^^ 

6.  It  is  required  to  find  the  cube  root  of  t\/-^' 

Ans. -/|,oriv'(3«). 

7.  It  is  required  to  find  the  square  root  of  x^—Ax^g 
'T-4a.  Ans.  x—2^a. 

8.  It  is  required  to  find  the  square  root  of  a-\'2^ab-\-b. 

Ans.  v^a-j-  ^  b. 


*  The  nth  root  of  the  m  power  of  any  number  a,  or  the  7nth  power  of 
m 
the  nth  root  of  a,  is  an. 
Also,  the  mn  root  of  the  mth  root  of  any  number  a,  or  the  mth  root  of  the 
1 
mn 
nth  root  of  a,  is  a 

From  which  ia?i  expression,  it  appears,  that  the  square  root  of  the  square 
root  of  a  is  the  4h  root  of  a  ;  and  that  the  cube  root  of  the  square  root  of  a. 
-T^he,  square  rto'  of  the  cube  root  of  a,  is  the  6th  root  of  «  ;  and  so  on  fc 
tbe  founb,  fifth,  or  any  other  numerical  root  of  this  kind. 


[BHATIONAL  QUANTITIES,  oa  SURDS,    71 


CASE  X. 

To  transform  a  hinomial^  or  a  residual  surd,  into  a 
general  surd, 

RULE. 


Involve  the  given  binomial,  or  residual,  to  a  power  cor- 
responding with  that  denoted  by  the  surd  ;  then  set  the 
radical  sign  of  the  same  root  over  it,  and  it  will  be  the  ge« 
neral  surd  required. 


EXAMPLES. 

1.  It  is  required  to  reduce  2-f-v'3  to  a  general  surd. 
Here  (2+ y^.S)"  =  4+3+4^/3=7+4^/  3  ;    therefore 

2+^^3  =  ^7+4^^.^,  the  answer. 

2.  It  is  required  to  reduce  ^2+y^3  to  a  general  surd, 

Here  (v'2+^/^=2  +  3+2v6=5+2^6 ; 
therefore^2+^3~^5+2^6j  the  answ^er. 

3.  It  is  required  to  leduce  ^2  +  ^4  to  a  general  surd. 
Here  (^2  +  3/4^ ^(J+t>^2  +  6y4  ;  therefore  3/2+ 

y 4  =  3/6 ( ]  +  V^V-O >  the  answer. 

4.  It  is  required  to  reduce  3—^/5  to  a  general  surd. 

Ans.  v^(14— 6^5). 

5.  It  is  required  to  reduce  \/2 — 2^6  to  a  general  surd» 

Ans.  ^(^26— 4;/ 12). 
5.  It  is  required  to  reduce  4 — ^7  to  a  general  surd. 

Ans.  v/(23— 8v'7). 
^.  It  is  required  to  reduce  2^3 — 3^9  to  a  general  surd, 
Ans.  1^/(1623/9—108^3—219.) 


72    IRRATIONAI.  QUANTITIES,  or  SURDg 


CASE  XI. 

To  extract  the  square  root  of  a  binomial^  or  residual  surd- 

RULK.* 

Substitute  the  numbers,  or  part;*,  of  which  the  given  surd 
is  composed,  in  the  place  ot'  the  letters,  in  one  of  the  two 
following  formulae,  arc»>rdmg  at,  i\  is  a  binomial  or  a  resi- 
dual, and  It  will  give  the  looi  required. 

And  if  the  second  part  of  the  binomi.il,  or  residual,  in 
this  case,  be  an  imaginary  surd,  the  same  theorems  will 
still  hold,  by  only  ciianging  -b  into  +6,  as  below. 

y'(a~^— 6)=y(i-f..v/(a=+6;;-v^(ia~i^(a»+6)) 

*  Prop.  1.  The  square  root  of  a  quantity  cannot  be  partly  rational  and 
partly  a  quadratic  surd.  If  possible,  lei  -^/  n  ~  a-^-'y/m  ;  then  by  squaring 

both  sides,  n  =  a    -|-2a^7n  m.  and  by  transposirion,  2a\/m  —  n-a^— m; 

therefore  y/  m  =  __f__JIL,  a  rational  quantity,  which  is  contrary  to  the 

2a 
supposition.    A  quantity  of  the  formv  o,  is  cfilled  a  quadratic  surd. 

Prop.  2.  In  any  equation  x -f -x/ v  =a-{-v/*5,  consisting  of  rational 
quantities  and  quadratic  si'rds,  the  rational  parts  on  each  side  are  equal, 
and  also  the  irrational  parts. 

If  X  be  not  equal  to  a.  let  a;  =  a  ^m  ;  then  a  .^m-^  ^/y  =  a-^y/  b,  or 
m-l- V  y<=  ^b  ;  that  ^b  is  partly  rational  and  partly  a  quadratic  surd, 
which  is  impossible,  (Prop  1.) ; 

,-.x=a,  and  y/y  ^^.y/bb. 

In  like  manner  if  a: — y/y-^a — >^  h  ;  then  a;  =  a,  and— v'y^  —  v'^- 

Prep.  3.  If  two  quadratic  ivrds  y/  x  and  v'  y-,  cannot  be  reduced  to 
others  which  have  the  same  irrntional  part,  their  product  is  irrational. 

If  possible,  let  y/ xy  =  rx,  where  r  is  a  whole  number  or  a  fraction: 
Then  xy  Si  ri  xs  ^  andy=r2ar;  .-.  y/y-^^  ry/  x:  that  is,  v'yandv^a 
may  be  so  reduced  as  to  have  the  same  irrational  part,  which  is  contrary  to 
the  supposition. 

Prop  4.  One  quadratic  surd,  ^  x,  cannot  be  made  up  of  two  others. 
v'  ni  andy/  n,  vjhich  have  not  the  same  irrational  part. 

If  possible,  let  y/  x  =  V  m-\"y''  n;  then  by  squaring  both  sides,  .t  =r 
'\-iy  mn-\.n,  and  x — m. — nc=8i/fnn,  a  rational  quantity  equal  to 
irrationtl,  which  is  absurd. 


IRRATIONAL  QUANTITIES,  or  SURDS.      73 

Where  it  is  to  be  observed,  that  the  only  cases  that  are 
useful  in  this  extraction,  are  when  a  is  rational,  and  a^—t 
in  the  first  of  these  formulae,  or  a^  -{-b  in  the  latter,  is  a 
complete  square. 

EXAMPLES. 

1.  It  is  required  to  find  the  square  root  of  1 1  -\-^12y  or 
Here, 


and 


^la~iya^~.6=  ^  ii—iy  121— 72=v^V--i=V'2 
Whence  ^(ll+6v/2)=34-y2,  the  answer  required. 
2.  It  is  required  to  find  the  square  root  of  3— 2^2. 
Ilere. 


^ig-fiv/g^— &=:v/|+iv/9--8=^/i-~i=V^2  ;   and 


Prop.  5.  The  square  root  of  a  binomial,  one  of  whose  terms  is  a  quadratic 
mrd,  and  the  other  rational,  may  sometimes  be  expressed  by  a  binomial,  one 
or  both  of  whose  terms  are  quadratic  surds. 

Let  a-^y/  b  be  the  given  binomial,  and  suppose  \/(a  -f-  V^i)  =a;4-y; 
vrhere  a?  and  y  are  one  or  both  quadratic  surds ;  then,  (see  Ryan's  Elementa- 
ry Treatise  on  Algebra,  Art.  367,)  ^{a~\^b)  esx—y,  .-.  by  multiplica- 
tion, y/  {a^—b)  =  x^— 2/3  ^ 

Vlso,  by  squaring  both  sidesof  the  first  equation,  a-\-\^bssx^  -|.2»y-f.y', 
and  (Prop.  2.)  .•:  a  ^  x"  •{•  y^ . 
Hence  by  addition,  c-f-  v'(a^ — ^)  =»  2x* , 
and  by  subtraction,  a— v'  («^  — ^)  =  %*  i 
.  TherootX'{-y^^\ia+i^(a^-b)\  -f-^  |ia_iV(a3-6)} 

From  this  conclusion  it  appears,  that  the  square  root  of  a^s^b  can  only 
be  expressed  by  a  binomial  of  the  form  cc-^-y,  one  or  both  of  which  are  quad- 
ratic surds,  when  a^ — b  is  a  perfect  square. 
i^  By  a  similar  process  it  might  be  shown  that  the  square  root  of  a— ^6,  or 

subject  to  the  same  limitation.  Ed. 

II 


74    IRRATIONAL  QUANTITIES  or  SURDS. 

Whence  ^{3 — 2^2) =^2 — 1,  the  answer  required. 

3.  It  is  required  to  find  the  square  root  of  6±2-x/5. 

Ans.  ^5±L 

4.  It  is  required  to  find  the  square  root  of  23^:8-^7. 

Ans.  4±V'7. 

5.  It  is  required  to  find  the  square  root  of  36  ±:  10 
^11.  Ans.  5±V(11). 

6.  It  is  required  to  find  the  square  root  of  33±:12^6. 

Abs.  3±2v^6. 

7.  It  is  required  to  find  the  square  root  of  1+4^^ — 3y 
or  l+y/— 48.  Ans.  2+V— 3. 

8.  It  is  required  to  find  the  square  root  of  3±4-y/— 1, 
or3±y'— 16.  Ans.  2±-v/-l. 

9.  It  is  required  to  find  the  square  root  of  —  1  -{-^  —  8. 

Ans.  l+\/-2. 

10.  It  is  required  to  find  the  square  root  of  a^-\-2xy/ 
(a2— a;2).  Ans.  x-\-^{a^-^x^), 

11.  It  is  required  to  find  the  square  root  of  6-f  2v^2— - 
^/(12)— /(24).  Ans.  l+v'2---v/3. 

For  Trinomial,  Quadrinomial  Surds,  &c. 

Rule.  Divide  half  the  product  of  any  two  radicals  by 
a  third,  gives  the  square  of  one  radical  part  of  the  root  j 
this  repeated  with  different  quantities,  will  give  the  squares 
of  all  the  parts  of  the  root,  to  be  connected  by  -f-  and  — . 
But  if  any  quantity  occur  oftener  than  once,  it  must  be 
taken  but  ©nee. 

For  if  x-{-y-\'2  be  any  trinomial  surd,  its  square  will  be 
x^  -{- y^  -^  z^  4-  ^xy  +  2x^  -{-  2y2  ;  then  if  half  the  pro- 
duct of  any  two  rectangles  as  2xi/-f-2x2'  (or  2x^yz)  be  di- 

2x^vz 
vided  by  some  third  2yZf  the  quotient  =x^,     must 

needs  be  the  square  of  one  of  the  parts  ;  and  the  like  for 
the  rest. 

EXAMPLE  I. 

To  extract  the  square  root  of  10  +  <•  (24)  +  ^  (40j 
4-  ^/  (60).  ^ 


IRRATIONAL  QUANTITIES,  oe  SURDS.    75 

Here  ^  ^^^^  "^  ^  (*°^-2  and  5^L(?1L^-^(!£L= 
""^        2V(^  '  2  V'  (40) 

^9  =  3,  and  ^°Ii^^^^=V'(25)=5.  And  the 
root  is  -v/2+  v/3+-v/5- 

EXAMPLE  2. 

It  is  required  to  find  the  square  root  of  12-{-y^(32)  — 
^  (48)4-/(80)— v'C"^4)+v'(40>-v/ (60). 

Here^^-^"— — ,— = — --,  this  produces  nothing. 
2  /  (8U)  ^5    '  *^  ^ 

./oKN     «        ^•v/(32X40)        ^^     _       ^(48X24) 
=  /  (25)=5 ;  and  ^^^-^=^4=2;  and-^-^— - 

=/9=3;  and  ^-g^^=  V  (16)  =  4,  &c.,  there- 

fore  the  parts  of  the  root  are  -v/4,  /5,  ^3,  v^2,  ^4,  djc, 
and  the  root  of  24-v'2 —  ^/^-^y/b ;  for,  being  squared,  it 
produces  the  surd  quantity  given. 

CASE  XII. 

To  extract  any  root  (c)  of  a  binomial  surd, 

RULE  I.* 

Let  the  quantity  be  A±B,  whereof  A  is  the  greater 
pr.rt  and  c  the  exponent   of  the  root  required.     Seek  the 

*  Let  the  sum  or  difference  of  two  quantities  x  and  y  be  raised  to  apower 
whose  exponent  isc,  and  let  the  \st^  3d,  bth,  7  th.  Sfc.  terms  of  that  poiver,  col- 
lected into  one  sum,  be  called  A,  and  the  rest  of  the  terms,  in  the  even  places, 
called  R  ;  the  dyff'erence  of  the  sq^iares  of  A  and  B  shall  be  equal  to  the  dif- 
ference of  the  squares  of  x  and  y  raised  to  the  same  power  c. 

For  ihe  terms  in  the  c  power  of  x-f-y,  writing  for  their  coefficients,  re- 

c         Q \  g 2  ^ 3 

spectively,  1,  c,  d,  e,  &,c.,  are  a;  -\-cx  -|-  dx        i/2'j-ex        y3-t~&c. 

1=  A-\-  B  ;  and  the  same  power  of  a — y  (changing  the  signs  in  the  even 

(»  Q 1  ^ ,Q  r 3 

places)  is  a; — ex        y-4-dx        y2 — ex        y^ '\- Sz^c.  =A—-B. 

And,  therefore,  (x-{-i/)  {x—y)  or(x2^yi)  =3(A  -f-  JB)  (A — B)>=^ 
J.3-£2.  q.E.D. 


•76    IRRATIONAL  QUANTITIES,  or  SURDS. 

least  number  n  whose  power  n"  is  divisible  by  A^ — B^,  the 
quotient  being  Q  compute  ^(A-f-BXv^Q)  in  the  nearest 
integer  number,  which  suppose  to  be  r.  Divide  Ay^Q 
by  its  greatest  divisor,  and  let  the  quotient  be  s,  and  let 


2^=t,    the    nearest   integer.        Then     the     root   = 


VQ 


-,  if  the  c  root  of  A±B  can  be  extracted. 


Let  one,  or  both  of  the  quantities,  x,y,be  a  quadratic  surd^  that  is,  let 
«+y»  the  c  root  of  the  proposed  binomial  A-{-£  belong  to  one  of  these 
forms,  p -hi Vq,ky^p-i  q,  or  k'^p  4  l^q.     And  it  follows  that, 

1.  If  a:-f-yc=:;j-f/^<7,  c  beirig  tuiy  whole  number,  A,  the  sura  of  the  odd 
terms,  will  be  a  rational  number ;  and  jB,  the  sum  of  the  terms  in  the  even 
places,  each  of  which  involves  an  odd  power  of  y,  will  be  a  rational  number 
multiplied  mto  the  quadratic  surd  y/ q. 

2.  Let  c,  the  exponent  of  the  root  sought,  be  an  odd  number,  as  we  may 
always  suppose  it,  because  if  it  is  even,  it  may  be  halvtd  by  the  extraction 
of  the  square  root,  till  it  becomes  odd  ;  and  let  x-j-3/=  k-s/pJ^q^.  Then  A 
will  involve  th«  surd  y/p,  and  B  will  be  rational. 

3.  But  if  both  members  of  the  root  are  irrational,  (ar-t-y  saifc^jo-f-^-Zj) 
A  and  B  are  both  irratif)nal,  the  one  involving  v'/'i  and  the  other  the  surd 
V9-  And  in  all  these  cases,  it  is  easih  seen  that  when  x  is  greater  than  y 
A  will  be  greater  than  B.  From  this  composition  of  the  binomial  A  +  B, 
we  are  led  to  its  resolution,  a&  in  the  above  rule,  by  these  steps. 

VfhenA  is  rational,  and  A2 — B2  is  a  perfect  c  power. 

1.  By  the  theorem  just  demonstrated,  A^—Bs  =^(x2  —ys)^  accurately  ; 
and  therefore  extracting  the  c  root  oi  A2 — JSa  it  will  be  x2 — y2  ;  call  this 
root  n. 

2.  Extract  in  the  nearest  integer,  the  c  root  of  ^  +JB,  it  will  be  (nearly' 
a?-f  ?/ ;  which  put  =  r.  ' 

3.  Dividers— 3/2  (=n)  by  z-f  v(=r)  the  quotient  is  (nearly)  x—y  \ 
and  the  sum  of  the  divisor  and  quotient  is  (rnore  nearly)  2x  \  that  is,  if  an  in- 

leger  value  of  x  is  to  be  found,  it  will  be  the  nearest  to    +r  * 
4.    x^—{x'^—y^)^y^;  or,  |r-4.^  |  ^  —ntsy^  :  whence 

V  =a  >W   (  ^-^-^j    — n  '•>  and  therefore,  putting  tz:sn-\-r    the  root  sough i 

x^y  —  i-^y/(X2 — n)\  the  same  expression  as  in  the  rule,  when  Qst\, 
s^=  i  ;  that  is,  when  A^  — J53  is  a  perfect  c  power,  and  the  greater  meniberj 
A  is  rational. 


IRRATIONAL  QUANTITIES,  or  SURDS.    77 

It  is  proper  to  observe  that  this  rule,  which  was  first 
given  by  Newton  in  the  Universal  Arithmetic ^  fails  when 
t=^^  exactly ;  in  which  case,  instead  of  taking  t  the  near- 

r-f— 

T 

est  integer  value  of  — —  >  it  must  be  taken  equal  to  \ : 

See  Ryan's  key  to  the  second  New-York  edition  of  Bon* 
nycastie's  Algebra. 


When  A  is  irrational,  and  Q=  1.    By  the  same  process,  x  =     T^- 

(  =  T)  andysa-y/  (Ts— n).  But  seeing  A  is  supposed  irrational,  and  c 
an  odd  number,  x  will  be  irrational  likewise  :  and  they  will  both  involve  the 
same  irreducible  surd  ^p,  or  s,  which  is  found  by  dividing'  A  by  its  greatest 
rational  divisor.  Write,  therefore,  for  x  or  T,  its  value  i)<,s,  and  x-|-  y  » 
f  s  +  V(fa  52— n). 

III. 

If  the  c  root  of  Aa — Ba  cannot  be  taken,  multiply  A2— Ba  by  a  nam' 
ber  Q,  such  that  the  product  may  be  the  {least)  perfect  c  power  Wl^-AsQ, 
~B2Q).  And  (now  instead  of  A  •+  B)  extract  the  c  root  of  (A-f-B)X\/Q, 
which  found  as  above,  will  be  t  5  -l^-y^  (fa  «2 — n)  ;  and  consequently  the 
c  root  of  A+B  will  be  t  a-f-V'  ^'^  52— n),  divided  by  the  c  root  of  V"  ^ ; 

that  IS, —^—pi ■* 

y^ 

In  the  operation,  it  is  required  to  find  a  number  Q,  such,  that(A2— -Ba) 
XQ  rnay  be  a  perfect  c  power  .  this  will  be  the  case,  if  Q  be  taken  equal 

g \ 

(0  (Aa — B2)  ;  but  to  find  a  less  number  which  will  answer  this  condi- 
tion, let  Aa— B2  be  divisible  by  o,  a, (m);  6,  6, (n) ;  d, 

wi      n    r  "• 

«, tf)\  &c.  in  succession,  that  is,  let  A  a.— B2  =«      h   d  &-c.  also 

IetQ=aa6  a  &c.  (Aa— Ba)   X  Q=sa  ^  X6  y^d         &,c. 

which  is  a  perfect  cth  power,  of  x,  y,z,  &c.  be  so  assumed  that  m-|»x,n+y, 
r-j*2;,  are  respectively  equal  to  c,  or  some  multiple  of  c.  Thus  to  find  a 
number  which  multiplied  by  180  will  produce  a  perfect  cube,  divide  180  as 
often  as  possible  by  2,  3,  b\  &c.  and  it  appears  that  2.  2.  3.  3.  5  =  180;  if, 

3      3      3  3 

therefore,  it  be  multiplied  by  2 .  3  .  5 .  5,  it  becomes  3 .  3  .  5  ,  or  (2  .  3 .  5)  , 
a  perfect  cube. 

If  A  and  B  be  divided  by  their  greatest  common  measure,  either  integer 
cr  quadratic  surd,  in  all  cases  where  the  clh  root  can  be  obtained  by  this 
method,  Q  will  either  be  unity,  or  some  power  of  2,  less  than  3c. 

If  the  residual  A— B  be  given,  it  is  evident  from  its  genesia  by  involution; 

h2 


78   IRRATIONAL  QUANTITIES,  or  SURDS. 

EXAMPLE. 

What  is  the  cube  root  of  v^  968  +  23. 

We  have  A^— B^  =  343  =  7  X  7  X  7.     Q  X  7'=n^ 

whence  n=7,  and  Q  =  1.     Then  V(A4-B  X  ^Z  Q)  = 

3/  56  -f  =  r  =  4.     A  ^  Q  =  v'SeS  =  22  -v/  2,  and  the 

r+-     4-}-^ 
radical  part  ^2=s,  and  — —'■=——;=    i    =    2,    in   the 
^       ^  2s       2^2 

nearest  integer.  And  ts=2^2y  ^(tV— n)  =  -vZ(8— 7) 
=  1.  V(Q=^-  ^nd  the  root  is  ?-^^-i^=  2  ^^  2 
-f-I,  whose  cube,  upon  trial,  I  find  to  be  ^968+25. 

RULE  II.* 

Let  the  surd,  that  is  to  have  its  root  extracted,  be  of 
the  form  1/  (a-|-^6),  or  \/{a—^b).     Then  if  a'—b  be  a 

that  the  same  rule  gives  its  root  x — y.  See  Universal  Arithmetic,  p.  139. 
Dr.  Waring's  Med.  Alg.  p.  287,  or  Maclaurin's  Aig.  p.  124. 

*  Thus,  let 'i/(a4- \/fc)«=  x-f-  v'y;  and  we  shall  have  by  involution. 

An  equation,  which,  by  expanding  the  right  hand  member,  and  comparing 
the  rational  and  irrational  parts,  gives 

n  ,  n(n— 1)  n— 2     ,  w(n— l)(7i— 2)(n— 3)  n— 4  2 

aszx    H 2—  *        y-i 2~ri ""        ^   "*" 

n— 1  n(/i— l)(rt— ?)  n—3     ^     ,  .     . 

Or,  which  is  the  same  thing,  under  a  different  form, 

V^6*,T  \{x-\->/yf''-(x^y/yf'Y 
Whence  by  squaring  each  of  these  equations,  and  subtracting  the  lalv 
from  the  former,  we  shall  have 

-\  \  (x+  V  yf  "*-  2  (ap=*  -so'^-i-C  a:—  ^yf  "  I  • 
Or,  by  rejecting  the  terras  that  destroy  each  other,  and  then  rauItiplyiDj. 

a"  -6^  (s^  ~2/)«,  or  x^  -y  =  (a^  -6^. 


IRRATIONAL  QUANTITIES,  or  SURDS.    1^9 

perfect  integral  cube,  and  some  whole  number,  can   be 
found,  that,  when  substituted  for  n,  will  make 

the  roots  of  the  two  expressions,  in  this  case  will  be 

And  if  the  second  part  of  the  binomial,  or  residual,  be  an 
imaginary  surd,  and  a^-l-6  be  a  perfect  integral  cube,  the 
extraction  may  be  effected,  by  finding  the  integral  value  of 
n  in  the  following  equation  as  before. 

In  which  last  case,  the  roots  of  the  two  expressions 
will  be,  

3/(a_^-.fe)  =  in— i  v/  ry— 43/a-4-6) 
each  of  which  formulae  may  be  obtained,  by  barely  changing 
the  sign  of  b  in  the  former. 

EXAMPLE. 

It  is  required  to  find  the  cube  root  of   10± 6-^/3,   or 
10±:-v/(10S). 

Here  a  =  10,  and  6  =  108  ;  whence  ^(a^— 6)   =  {/' 
(100— 108)=— 2,  and  w'— 3(3/^5:i.T)n=20, 

or  n^  -{-  Hn=20 
where   it  readily  appears  from  inspection,   that  w  =  2, 
Whence  ^  (10  4-  ^/  108)  =  2-f-i^(4— 4  X  —  2)  ~ 
l+^^/  (12)  =  1  +  ^3,  and  V(10— v/  1^8)  =  |— | 
^{4,^4:  X  —  2)  =  I— ^y  12  =:  1  -  v'S. 


Where,  supposing  a^ — b  to  be  a  complete  power  of  the  nth  degree,  let 
(a^—b)"'  be  put=ac. 

Then,  since  ac^ — 2/==Ci  »nd  consequently  3/«=a:*—c,  if  this  value  be  sub« 

,  -  .     ^  .        n  ,  n(n—1)  n— 2        n(n— l)(n-— 2)(n— 3) 

sntuted  for  2/,  in  the  equation  at    -f- — -x        y-f- --r ■ 

n— 4 
X         y^  -|-  Slc.  as  a,  we  shall  obtain  an  equation,  in  which  the  value  of  a', 
as  before  mentioned,  is  irrational,  when  the  extraction  required  is  possible, 
•See  Wood,  or  Ryan's  Algebra,  E». 


80   IRRATIONAL  QUANTITIES,  or  SUrSs. 


EXAMPLES  FOR  PRACTICE, 


L  Required  the  cube  root  of  68—^/4374. 

^"'-  "172- 

2.  Required  the  cube  root  of  11  -f  5  y/  7. 

3.  Required  the  cube  root  of  2A/7-i-3*/3. 

,       ^/7-i-v/3 

4.  Required  the  fifth  root  of  29^6+4  V3. 

5.  Required  the  cube  root  of  45 ±29v^2. 

Ans.  3+x/2,  and  3— </2. 

6.  Required  the  cube  root  of  9  ±4^5,  or  9±v^S0. 

Ans.  5  4-|y^5,  and  l—W^- 
1,  Required  the  cube  root  of  20^68^^ — 7. 

Ans.  5-h-^— 7,  and  5 — y/ — 7, 
6.  It  is  required  to  find  the  cube  root  of  35 ±69^ — 6. 

Ar.s.  o-{-  v/  —  ^>  ^"<^  ^ — v/ — ^* 
9.  It  is  required  to  find   the   cube  root  of  i!>\±.^'- 
2700.*  Ans.  — a+^y^— 3,  and  —3—2^3. 


*  Whenever  it  can  be  done,  the  operation,  in  cases  of  this  kind,  ought  to  be 
abridged,  by  dividing  the  given  binomial  by  the  greatest  cube  that  it  contains, 
and  then  finding  the  root  of  the  quotient ;  wh:ch  being  multiplied  by  the  root 
of  the  cube,  by  which  the  bino.iiial  was  div  de'i,  will  give  the  root  required. 
Thus  in  the  example  above  given,  8]  +  ^—2700  =  27X  C3+  V  —  ^^) 
where  the  root  ofx-{-*/ — ^-^'  being  now  more  easily  found  to  be— l+Sy' 

'~3,  —14- 3  V  —3,  we  shall  have  by  multiplying  by  3,  (which  is  the  cube 

root  of  27), -323 V —3,  as  above. 


IRRATIONAL  QUANTITIES,  or  SURDS,    81 


CASE  XIII. 


To  find  such  a  muliipUfr,  or  multipliers  y  as  will  make  an^ 
*    binomial  surd  rational. 


RULE.; 


1.  When  one  or  both  of  the  terms  are  any  even  roots^ 
multiply  the  given  binomial  <»r  residual,  by  the  same  ejc- 


Also  this  is  useful,  in  Cardan's  rule  Tor  cubic  equations  ;  thus,  3/(81  -f-  V 
(— 2700)) -f-^(81-.^(- 2700))-= —3x2=  —6.  or  =.— |  X  2  =—3,   or 

•-X  2=3  9,  the  imaginary  parts  vanist-H.<r,  by  the  contrariety  of  their  signs. 
See  De  Moivre's  appendix.  toSanderso^i's  Algebra,  Universal  Arithmetic,  or 
Maclaurin's  Algebra. 

*  If  a  multiplier  be  required,  that  sbf.il  render  any  binoBnial  surd,  whether 
it  consist  of  even  or  odd  roots,  rations',  it  may  be  found  by  substituting  the 
given  numbers,  or  letcers,  of  which  ir  is  composed,  in  the  places  of  their 
equals,  in  the  following  general  formuU; 

Binomial  \/a  -i-  y/b. 

Multiplier  y a"~^  ^Z y  a^~\  +  ^a'^~^h ^  T !J/«''"~^i  ^  +  &c; 

where  the  upper  sign  of  the  multiplier  ntust  be  taken  with  the  upper  sign  of 
the  binomial,  and  the  lower  with  the  Ir.wer  ;  and  the  series  continued  to  U 
terms. 

This  multiplier  may  be  derived  from  observing  the  quotient  which  arises; 
from  the  actual  divisioQ  of  the  numerator  by  the  denominp.tor  of  the  follow 
ing  fractions :  thus, 


S2   IRRATIONAL  QUANTITIES,  or  SURDS. 

pression,  with  the  sign  of  one  of  its  terms  changed  ;  and 
repeat  the  operation  in  the  same  way,  as  long  as  there  are 
surds,  when  the  last  result  will  be  rational. 


-  a  —V           n— 1  .     n— 2        n— 3  2  n— 1  ^ 

I.      ^    6=  X        -hx       y-^x        y  -J-  &c +3/         ton 

terms,  whether  n  be  ewen  or  odd. 
n      n 
__  a  —y           n—l      n— 5j     ,     n— 3  a  n— 1    ^ 

II. -^^=zx       — X       y-f-x       y  — &.C — y  to    n 

terms,  where  n  is  an  even  number. 

TTT  *    -f-  V**        n—l      n— 2     .     n— 3  2       „  ^    .     n—l 

lil._ — !~£— =2/       —X        y-f-x       y  —  fcc "Ty         to  n 

a+  y 
terms,  when  n  is  an  odd  number. 

Now  let  X    eaa,y    =a6;  then  x=»>/a,  y  =  v  *?  and  these  fractions 

.everan,  become  ^^,  -^^^,  .„a  ~^„^- 

And.smcex  =aVa         ,   a:         «=  V  a         ,  &c.  also  y**  c=  V^   , 

3        "/,3 
y     =.V  6    ,  &,c.  therefore, 

,-^^=  Va"-'  +v  a"-^6  +  Va"-'6=  +  to. . .  +^6"-' 

V  a — ^6 

to  n  terms  ;  where  n  may  be  any  whole  number  whatever.    And, 

V '^-f-V  "  « j 

to  n  terms  ;  where  the  terms  b  and  v^A  have  the  sign  +,  when  n  \? 

an  odd  number;  and  thp  si'^n  — ,  when  n  is  an  even  number. 

Now,  since  the  dhnsor  multiplied  by  the  quotient  gives  the  dividend,  it  ap- 
pears from  the  foregoing  operations  that,  if  a  binomial  surd  of  the  form 

Va^Vb  he  multiplied  by  V ^~^     V  J'-'^b  4-  &c.  .  .  .  +V  b^-^\ 
(n  being  any  whole  number  whatever),  the  product  will  be  a — 6,  a  rational 

quantity  ;  and  if  a  binomial  surd  of  the  form  V  a-f-V  6  be  multiplied  by 

y  o         —y/a  b-^y/a         b   —  &,c.  .   .  •^^  V  ^         ,   the  product 

will  be  a-\'b,  or  a— 5  ;  according  as  the  index  n  is  an  odd  or  an  even  num 
ber ;   See  my  Elementary  Treatise  on  Algebra,  Theoretical  and  Practical. 


IRRATIONAU  QUANTITIES,  or  SURDS.    8§ 

2.  When  the  terms  of  the  binomial  surd  are  odd  roots, 
Ihe  rule  becomes  more  complicated  ;  but  for  the  sum  or 
difference  of  two  cube  roots,  which  is  one  of  the  most  use- 
ful cases,  the  multiplier  will  be  a  trinomial  surd,  consisting 
of  the  squares  of  the  two  given  terms  and  their  product, 
with  its  sign  changed. 

EXAMPLES. 

1.  To  find  a  multiplier  that  shall  render  5-f-v/3  ra- 
tional. 

Given  surd  64-^3 
Multiplier  6— ^^3 

Product  25—3=22,  as  required. 

2.  To  find  a  Multiplier  that  shall  make  V  5-{-'v/3  ra- 
tional. 

Given  surd  ^5 +-^3 
Multiplier  y/o  —  ^3 


Product         5  —  3=2,  as  required. 

3.  To  find  multipliers  that  shall  make  yS-f-yS  ra* 
tional. 

Given  surd     ys+ys 
1st  multiplier  ys-ys 

1st  product      ^5 — ^3 
2d  multiplier   y^5-f  ^3 

2d  product         5  —  3=2,  as  required. 

4.  To  find  a  multiplier  that  shall  make  yT-fV'^  i'^- 
tional. 


^4    IRRATIONAL  QUANTITIES,  or  SURDS= 


Given  surd  yr+ys 

Multiplier    ^/T^— ^(7  X  3)  +  y33 

7-l-y(3X72) 

— V(3X7-j-y(7X32) 

4-^(7X3^)4-3 

Product       7+3=10,  as  was  required. 

5.  To  find  a  multiplier  that  shall  make  -v/  5  —  v^  a:  ra- 
tional. Ans.  ^5+\/x, 

6.  To  find  a  multiplier  that  shall  make  ^  a  *f  -v/  6  ra- 
tional. Ans.  ^a — v^^' 

7.  To   find  multipliers  that  shall  make  a  +  \/  b  ra- 
tional.  Ans.  a — ^h. 

8.  It  is  required  to   find  a  multiplier  that  shall  make 
l—2/2a  rational.  Ans.  l+V2a+y4a2. 

9.  It  is  required  to  find  a  multiplier  that  shall  make 
V^—W^  rational.  Ans.  ^0+1^6+1^4. 

10.  It  is  required  to  find  a  multiplier  that  shall  make 
i/(a')4- i/(&'),  or  af +6f  rational. 

Ans.  ^a^—\aPb^)-\'\/{a%^)--\/h\ 

CASE  XIV. 


To  reduce  a  fraction,  whose  denominator  is  either  a  simple 
or  a  compound  surd,  to  another  that  shall  have  a  rational 
denominator, 

RULE. 

1 .  When  any  simple   fraction  is  of  the  form  — r-,  rauU 
tiply  each  of  its  terms  by  ^a,  and  the  resulting  fraction 

will  be  ^". 

a 


iRRATIOKAL  QUANTITIES,  or  SURDS.     85 

X)r  when  it  is  of  the  form  — -,  multiply  them  by    \/c?i 

and  the  result  will  be  ~- — . 
a 

h 
And  for  the  general  form  — -,  multiply  by  ^a"— ^,  and 

Ine  result  will  be  -^ ^ 

a 

2.  If  it  be  a  compound  surd,  find  such  a  multiplier,  by 

the  last  rule,  as  will  make  the  denominator  rational ;  and 

multiply  both  the  nu  nerator   and    denominator   by  it,  and 

the  result  will  be  the  fraction  required. 

EXAMPLES. 

2  3 

1.  Reduce   the   fractions  — ^  and ——-,  to  others  that 

V  "         ^v  ^ 
shall  have  rational  denominators. 

^  ^   — •— ^ — =-5/125  the  answer  required. 
1X5  5  5 

Q 

2.  Reduce  — — to  a  fraction  whose  denominator 

shall  be  rational. 

3  v/5+v/2_3^5+3v/2^3v/5-f  3yg 

^'^^'^^  V'^— ^2^v^5  +  v'2  5-2  3 

=:^^^-, — -^^  =  v'S+v/S  the  answer  required. 

4/2 

3.  Reduce—- — 5  to   a  fraction,   whose  denominator 

shall  be  rational. 

R  \/2    _     v/2x(3-fv/2)       _3^2+2^2-f3v/2 

3-v^2~(3-v'2)X(3+^2)        9-2  7 

2     3 
==+r\/2  the  answer  require/!. 


86    IRRATIONAL  QUANTITIES,  or  SURDS.      " 

4.  Reduce  — -^ — —  to  a  fraction,  that  shall  have  a  ra- 
tional  denominator. 

Ans.  v^(^a)-^/('8), 
4 

5.  Reduce  —-. to  a  fraction  that  shall  have  a  ration- 

al  denominator. 

-*"=•  -T---7- 

6.  Reduce  — ; — -  to  a   fraction,   the   denominotor     of 

which  shall  be  rational.  ^    Ans. ;; — --^-, 

a-~6 

7.  Reduce—— — —— to  a  fraction  that  shall  have  a  ra- 

tional  denominator. 

Ans.  5X(V'(49)+V'(35)  +  y(25)}. 

3/3 

S.  Reduce  -r—i  ^tt-k  to  a  fraction  that  shall  have  a  ra- 
tional  denominator. 

19 
4 

9.  Reduce  t-tttit^  to  a  fraction  that  shall  have  a  ra- 

tidnal  denominator. 

Ans.  4^  '-^10-2^2-{-(2i'^5)X{/o\ 


ARITHMETICAL  PROPORTION,  &c.        Sf7 

OF  • 

ARITHMETICAL  PROPORTION 

AND  PROGRESSION. 

Arithmetical  Proportion,  is  the  relation  which  two 
quantities  of  the  same  kind,  have  to  two  others,  when  the 
difference  of  the  first  pair  is  equal  to  that  of  the  second. 

Hence,  three  quantities  are  said  to  be  in  arithmetical 
proportion,  WMcn  the  difference  of  the  first  and  second  is 
equal  to  the  difference  of  the  second  and  third. 

Thus,  i!,  4,  6,  and  'i,  a-{-b,  a4- :b,  are  quantities  in  arith- 
metical proportion. 

And  four  quantities  are  said  to  be  in  arithmetical  propor- 
tion, when  the  difference  of  the  first  and  second  is  equal 
to  the  difference  of  the  third  and  fourth. 

Thus,  3  7,  12,  16,  and  a,  a-i-6,  c,  c-f^j  are  quantities 
in  arithmetical  proportion. 

Arithmetical  Progression  is  when  a  series  of  quan- 
tities increase  or  decrease  by  the  same  common  difter- 
ence. 

Thus,  1 ,  3,  o,  7,  9,  &c.  and  a,  a-[-d,  a-\-'id,  (X+3  /,  &c. 
are  increasing  series  in  arithmetical  progression,  the  com- 
mon differences  of  which  are  2  and  d. 

And  I  5,  12,  9,  (i,  &c.  and  a,  a — /,  a — 2d,  a  -  'Sd,  &c.  are 
decreasing  series  in  arithmetical  progression,  the  common 
differences  of  which  are  3  and  d. 

The  most  useful  properties  of  arithmetical  proportion 
and  progression  are  contained  in  the  following  theorems  : 

1.  If  four  quantities  are  in  arithmetical  proportion,  the 
sum  of  the  two  extremes  will  be  equal  to  the  sum  of  the 
two  means. 

Thus  if  the  proportionals  be  2,  5,  7,  10,  or  a,  fc,  c,  d  ^ 
then  will  2-f-10=5-{-7,  and  a-|-ci=64-c. 

2.  And  if  three  quantities  be  in  arithmetical  proper- 


88  ARITHMETICAL  PROPORTION 

lion,   the  sum  of  the   two   extremes   will  be  double  the 
mean. 

Thus,  if  the   proportionals  be   3,  6,  9,  or  a,  6,  c,  ther 
will  3+9=2  X 6=  1 2,  and  a-f  c=26. 

3,  Hence  an  arithmetical  mean  between  any  two  quan- 
tities is  equal  to  half  the  sum  of  those  quantities. 

Thus,  an  arithmetical  mean  between  2  and  4  is  =— ^ 

54-6 
—3 ;  and  between  5  and  6  it  is  =  ——-=64. 


And  an  arithmetical  mean  between  a  and  b  is  — —  .* 

4.  In  any  continued  arithmetical  progression,  the  sua^ 
of  the  two  extremes  is  equal  to  the  sum  of  any  two  terms 
that  are  equally  distant  from  them,  or  to  double  the  mid- 
dle term,  when  the  number  of  terras  is  odd. 

Thus,  if  the  series  be  2,  4,  6,  8,  lO,  then  will  2-j-lO— 
4-1-8  =  2x6=1^^. 

And,  if  the  series  be  a,  a-\-df  a-{-2fI,  a+'^(/,  a-^-^d, 
then  willa  +  (a-h4r/)  =  (a-fr^)4-(^'+3c/)=2X^a-fi:£^). 

5.  The  last  term  of  any  increa.^ing  arithmetical  series 
is  equal  to  the  first  term  plus  the  product  of  the  common 
difference  by  the  number  of  terms  less  one  ;  and  if  the 
series  be  decreasing,  it  will  be  equal  to  the  first  term  mi- 
nus that  product. 

Thus,  the  nth  term  of  the  series  a,  a-\-d,  a-r2dy 
a-^-Sd,  a-\-4d,  &c.  is  a+(n-  l)c/. 


*  If  two,  or  more  arUhmetical  means  between  any  two  quantities  be  re 
quired,  they  may  be  pxpressed  as  below. 

Thus,  ""■  7"  ■  and =  two  arithmetical  means  between  a  and  b,  c 

3  S 

being  the  less  extreme  and  b  the  greater. 

^     ,na-{-b   (n— l)o  +  26    (n—2)a-\-3b   ^  a  +  nb  ^      , 

And  — 7^-,  !: ^—~ — ,  ^ ^—^ — ,  &lc.  to  ——-=anv  number  (n 

Ti-fl'       714.1        '       n  +  l  n+1 

of  arithmetical  means  between  a  and  6  ;  where  — ;  is  the  common  diflfer 

ence  ;  which  being  added  to  a,  gives  the  first  of  these  naeaa?  5  and  the> 
again  to  this  last,  gives  the  second  ;  and  so  on 


AND  PROGRESSION.  89 

And  the  nth  term  of  the  series  a^a-^d,  a— 2d,  a— 3rfj 
a  — 4c?,  &c.  is  a  — (n—  \)d. 

6.  The  sum  of  any  series  of  quantities  in  arithmetical 
progression  is  equal  to  the  sum  of  the  two  extremes  muK 
tipHed  by  half  the  number  of  terms. 

Thus,  the  sum  of  2,  4,  6,  8,  10,  12,   is  =  (2+12)  X 

6 

-=14X3=42. 

2 

And  if  the  series  be  a-f-(a-|-^)4-(a+2ri)-{-(a-{-3</)4- 
{a-\-4d)  &c.  .  .  .   -\-l,  and  its  sum  be  denoted  by  S,  we 

shall  have  S=(a+/)X^,   where  Hs  the  last  term,  and  n 

-*» 

the  number  of  terms. 

Or,  the  sum  of  any  increasing  arithmetical  series  may 
be  found,  without  considering  the  last  term,  by  addmg  the 
product  of  the  common  difference  by  the  number  of  terms 
less  one  to  twice  the  first  term,  and  then  multiplying  the 
result  by  half  the  number  of  terms. 

And,  if  the  serie*^  be  decreasing,  its  sum  will  be  found 
by  subtracting  ihe  above  product  fr  >m  twice  the  first  term, 
and  then  multiplying  the  result  by  half  the  number  of 
terms,  as  before. 

Thus,  if  the  series  be  a-{-{a-^d)-\-{a-\'2d)'^{a-\-3d) 
|-(a4-4(/),  &c.  continued  to  n  terms,  we  shall  have 

S=  j  2a-\'{n^ld)  I  X^. 

And  if  the  series  be  a+(a-i/)  +  ((i— 2ti)+(a-3fZ)-{- 
"rt — id),  &c.  to  n  terms,  we  shall  have 

5=J2a-(n-l)ri|  x|(*). 


(»)  The  sum  of  any  number  of  terms  (n)  of  the  series  of  natural  numbers 
V,2,3,4,5,6,7,&c.  13=^''-^^. 

Thus,  1 4.2-^3-{w4-{-  5,  &c.  continued  to  100  terms,  is= ^51 

/<101«5050 


90        ARITHMETICAL  PROPORTION,  4*. 


EXAMPLES. 

1.  The  first  term  of  an  increasing  arithmetical  series  it 

3,  the  common  difference  2,  and  the  number  of  terms  20  ;- 
required  the  sum  of  the  series. 

First,   3+2(20-1)— 3-f2X19=3-}-38=4I,    the   las: 

term. 

20  20 

And  (3-f  4 1)  X-^=44  X  — =44  X  10=440,  the  sum  re 

quired. 

Or,  {2X3-f-(20-l)X2]XY=(6-fJ9X2)X10=^(6 

f  38)  X  10=44X10=440,  as  before. 

2.  The  first  term  of  a  decreasing  arithmetical  series  is 
100,  the  common  difference  3,  and  the  number  of  terms^ 
34;  required  the  sum  of  the  series. 

First,  100— 3(34—1)=  100-3X33=^  100—99=1,  the 
'ast  term. 

And  (lOOf  Ox— =101  X  — =  101  X  I7  =  1717,    the 
2  2 

3Uin  required. 

Or,  {2X100— (34-1)  X3}Xy=(2C0— 33X3)  xr: 

^(200-99)x  17=101X17=1717,  as  before. 

3.  Required  the  sum    of  the    natural  numbers,  1,2,3. 

4,  5,  6,  &c.  continued  to  1000  terms.  Ans.  500500, 

4.  Required  the  sum  of  the  odd  numbers  1,  3,  5,  7,  d, 
&c.  continued  to  101  terms.  Ans.  1020!, 


Also  the  sum  of  any  number  of  terms  («)  of  the  series  of  odd  numbeit 
i,  3,  5,  7,  9,  n,  &c.  is  =n2. 

Thus,  1  4-3-1-5  -f  7-4-9,  <fec.  continued  (o  50  terms,  is  =350^  -s  2500. 

And  if  any  three  of  the  quantities  o,  d,  »,  S,  be  given,  the  fourth  may  bf 
found  from  the  equation 

Where  the  upper  sign  J^  is  to  be  used  when  the  series  is  increasing,  and  the 
lower  sign  —  when  it  is  decreasing;  also  the  lasUrrm  ?  =  a+(n— l)f?,  a*- 


GEOMETRICAL  PROPORTION.  Qi 

6,  How  many  strokes  do  the  clocks  of  Venice,  which 
goon  to  24  o*clock,  strike  in  a  day  ?  Ans.  300 

6.  Required  the  365th  term  of  the  series  of  even  num= 
bers  2,  4,  6,  8,  10,  12,  &c.  Ans.  730. 

7.  The  first  term  of  a  decreasing  arithmetical  series  is 

10,  the  common  difference  -,   and  the   number  of  terms 
o 

21  ;  required  the  sum  of  the  series.  Ans.   140. 

8.  One  hundred  stones  being  placed  on  the  ground,  in 
a  straight  line,  at  the  distance  of  a  yard  from  each  other  : 
how  far  will  a  person  travel,  who  shall  bring  them  one  by 
one,  to  a  basket,  placed  at  the  distance  of  a  yard  from  the 
^rst  stone  ?  Ans.  5  miles  and  1300  yards* 

OF 
GEOMETRICAL  PROPORTION 

AND 

PROGRESSION. 

^  Geometrical  Proportion,  is  the  relation  which  UVo 
quantities  of  the   same  kind  have  to  two  others,  when  the 


*  If  there  be  taken  any  four  proportionals,  a,  b,  c,  d,  which  it  has  beer- 
usual  to  express  by  means  of  points  :  thus, 
a  :  b  :  :  c  :  d, 

a      c 
:his  relation  will  be  denoted  by  the  equation  t  — -r"'   *^^'^^'"^  *''^  eqiiu!  ratios 

are  represented  by  fractions,  the  nuineratois  o{  which  are  the  antecedents, 
and  the  denominators  the  consequents.  Hence,  if  each  of  the  iv.o  ;nerr>bers 
of  this  equation  be  multiplied  by  bd,  there  will  anse  ad  =:bc.  From  which 
it  appears,  as  in.the  common  rule,  that  the  product  of  the  two  extremes  of 
any  four  proportionals  is  equal  to  that  of  the  means.  And  if  the  third  t,  in 
this  case,  be  the  same  as  the  second,  ore  =6,  the  proportion  is  said  to  be 
continued,  and  we  have  adt=,bi,  or  6s=  v/od;  where  it  is  evident,  that  the 
product  of  the  extremes  of  three  proportionals  is  equal  to  the  square  of  the 
mean;  or,  that  the  mean  is  equal  tal^e  square  root  of  the  prorftrct  of  the  (Wf 


32 


GEOMETRICAL  PROPORTION 


antecedents,  or  leading  terms  of  each  pair,  are  the  same 
parts  of  their  consequents,  or  the  consequents  of  the  ante- 
cedents. 


Also,  if  each  member  of  the  equation  ad  ==.6c  be  successively  divided  bv 
ti,  dc,  ac,  &c.  the  results  will  give 
a       c  " 


&c. 


d 
b_ 

*d 
d_ 


Or  the 


proportions 


h::c 
c::b 
a::d 
d:c. 


30  that,  by  following;  this  method,  we  can  easily  obtain  all  the  transformations 
of  the  terms  of  the  proportion,  that  can  be  made  to  agree  with  the  equations 
a  J=  be. 
In  like  manner,  from  the  same  eqoaMty  —=»-,  there  will  result,  by  multi- 

,  ,      .  .     ,       ,  ma      nc    ma     mc 

plication,  the  following  equivalent  forms  :  —  =—  ;  —  =  - ^  ; 

Which,  bping  converted  into  proportions,  become  Jtm  :  mb  :  :  nc  :  nd,  and 
ma  :  nb  :  :  mc  :  nd.     And,  by  lakinsr  any  like  powers,  or  roots,  of  the  differ- 

ent  sides  of  the  same  equation,  we  have  — 1=  — .     Or,  putting  the  terms 

in  the  form  of  a  proportion,  a^  :  b^  :  :  c^  :  d"^.     In  which  cases  m  and  Xi 
may  be  any  whole  or  fractional  numbers  whatever. 
Attain,  if  there  be  taken  the  several  equations 
c  "I 

which  correspond 


with 


f.f: 

i:k: 


:C:d 

I :  m 


b       d 

^     n 

\_z=  !l\     tbe  proportions 
km]  \ 

&c.  &c. 

vve  Shall  have,  by  multiplying  their  like  terms,  y^^^j^^  "r^X^&c 

Or  by  putting  the  expression  into  the  form  of  a  proportion,  aei  &c.  :  hfk 

Slc.  '.-.cglSic.:  dhm  &c.     Also,  taking  ^  =  j,as  before,  we  shall  have,  by 

multiplication,  ^^  ^Z,  and  by  augmenting  or  diminishing  each  side  oi 

nb      mc-i-nd        ,  .  ,.    , 
; —  ;   which,  be- 


nb       nd' 
ma 


the  equation  by  I ,  ^^ 

le  f( 
mc 

And  if  the  above-mentioned  equ^ 


mc  ma  ^^ 

~nd  "~     '  nb 


nd 


ing  expressed  in  the  form  of  a  proportion,  gives  ma^nb  :  nb  :  :  mc-±_nd . 
na\  or  ma-^nb  -.  mc^nd  -.  :  nb  -.  wdy^ 

|<  Jf  ==T»  ^^  P"t  by  a  similar  rauUi 


AND  PROGRESSION.  9- 

And  ii  two  quantities  only  are  to  be  compared  together, 
the  part  or  parts,  which  the  antecedent  ib  of  its  consequent^ 


plication  of  its  terms,  under  the  form^  =^  and  tr.en  augmented  or  dimi 

(JO      qa 
nished  by  1,  as  in  the  last  case,  there  will  arise  pa-^qb  -.  pc-^qd:  :  qb  :  qd, 
Whence,  dividing  each  of  the  antecedents  of  thebe  two  analogies  by  their 
.„    .     ma-\~nb      nb      b         ,pa-t~qb      qb 

consequents,  the  result  will  give — = — ,  =» — ,  =  -  ;  and =»— > 

°      mc-^nd      nd      d  pc-jrqd      qd 

=9  -.     And,  consequently,  as  the  two  right  hand  .nembers  of  these  expres- 

b  ,    ,.  ,         ma-i-nb      pa  -,-  qb 

sjons  are  each  =-„  we  shall  have  ._^=— -= ---=^-i-. 

d  mc±-nd     pc  ^  qd 

Or,  by  converting  the  corresponding  terms  of  ihis  equation  into  a  proper 
{ion  ma  +  n6  :  mc^nd  -.    pa-^^qb    pc-^qd.     Also,  because  the  common 
a      c  a      b 

^  =,- gives  ^:=.^, 

ma     mb        ,  pa     pb  .    „    u.  •      u  •     i 

—   =3 — ,,  and  —  a—-,  we  shall  obtam,  by  a  similar  process,  ma -t-nc  : 
nc       nd  qc      qd  t^  ♦        _ 

pa-±^qc  :  :  mb  ;^nd  :  pb  r^qd;  which  two  analogies  may  be  considered  as 

general  formulae  for  chatij^'ing  the  terms  of  the  propo:«ion  a  :  b  :  :  c  :  d,  w.th- 

out  altermg  its  nature.      Thus,  by  supposing  m,  n,  p,  y,  to  be  each  =  1,  and 

taking  the  antecedfuts  with  the  superior  signs,  and  the  consequents  with  the 

inlerior,  we  have  a-\-b  .  a-  b  -.     c-j-d  :  c — d,  and  a-^c  ;  a — c     :  6  ^  rf  r 

h — (/;  which  forms,    together  with  several  of  those   alr«:ady  given,  are   the 

usual  transformations  of  the  common  analogy  point,  d  out  above. 

In  like  manner,  by  takmg  m,  n  and  p  each  <=  1 ,  and  q  =0,  there  will  arise 
a  j^6  .  a  :  :  c  -t  d  :  c,  and  a-^c  :  a  :  :  b-^d  b  ;  each  of  w'lich  pioportions 
mav  be  verified  by  making  the  juoduct  of  the  extrem^-s  equal  to  that  of  the 
means,  and  observing  that  ad  =  be. 

Lastly,  takmg  any  number  of  equations  of  the  form  before  used,  for  ex 
«       c       e        g-        -  ,     ,  ,.  , 

pressmg   proportions,  as  —  =a  -  =  -=  — :=&c.  ;  which,  according  to  the 

b       d      J       n 
common  method,  are  called  a  series  of  equal  ratios,  and  ^re  usually  denoted 
by  a  :  6  :  :  c  :  d  :  :  e  -.f.  :  g  :  fi  :  :  &c.  we  shall  ne^  f-ssarily    have  from  the 

fractions  being  all  equal  to  each  other- =9, -r-  "=5,^  '^I'T  ^=  ??  ^-^ 

And  by  multiplying  q  bv  each  of  the  denominators,  a  =■  bq,  c  =:dq,  e  safq^ 
g-'^'^y,  &c. 

Whence,  equating  the  sum  of  all  the  terms  on  the  left  hand  side  of  these 
equations,  with  those  on  the  right,  we  havea-f-c-f-e-J-g--}-  &-£"  ^(6f.rf+ 
/'_j_/j-|_&c.)9.  And  consequently  by  division,  and  tlie  properties  of  propor 
tionals  beforf^^  shown, 

a-\-c-\-e-\-g-\-8j.c._a_^a-\-c_a^c-k-e 
b^-d-\-f  4-/1 -{-&c.  ^  6  ^  b'~J^d^  b~^^'^    '^' 
which  results  show,  that,  in  a  series  of  equal  ratios,  the  sum  of  any  nurabe: 
of  the  antecedents  is  to  that  of  their  consequents,  as  one,  or  more  of  the  ar 
'or-prients,  is  to  one,  or  the  same  number  of  consequents,     ^.  E.  D. 


94  GEOMETRICAL  PROPORTION 

or  the  consequent  of  the  antecedent,  is  called  the  ratio  ; 
observing,  in  both  cases,  always  to  follow  the  same  method. 

Hence,  three  quantities  are  said  to  be  in  geometrical 
proportion,  when  the  first  is  to  the  same  part,  or  multiple, 
of  the  second,  as  the  second  is  of  the  third. 

Thus,  3,  6,  12,  and  a,  ar,  ur^,  are  quantities  in  geome- 
trical proportion. 

And  four  quantities  are  said  to  be  in  geometrical  propor- 
tion, when  the  first  is  the  same  part,  or  multiple,  of  the 
second,  as  trie  third  is  of  the  fourth. 

Thus,  if  8,  3,  12,  and  a,  ar,  6,  6r,  are  geometrical  pro- 
portionals. 

Direct  proportion,  is  when  the  same  relation  subsists 
between  the  first  of  four  terms  and  the  second,  as  between 
the  third  and  fourth. 

Thus,  3,  6,  5,  It),  and  a,  ar,  6,  6r,  are  in  direct  propor- 
tion. 

Inverse,  or  reciprocal  proportion,  is  when  the  first  and 
second  of  four  quantities  are  directly  proportional  to  the 
reciprocals  of  the  third  and  fourth  : 

Thus,  2,  6,  9,  3,  and  u,  ar,  br,  6,  are  inversely  propor- 
tional ;   because   2,  b,  -,  -,  and  a,   ar,  ■;— >  r  ^^®  dii'ectly 
y   o  or    0 

proportional. 

GeoMtTRiCAL  Progrkssion  is  when  a  series  of  quan- 
tities have  the  same  conf-tant  ratio  ;  or  which  increase,  or 
decrease,  by  a  comm(»n  multiplier,  or  divisor. 

Thus,  ^,  4,  H,  16,  .h2  64,  &c.  and  o,  ar,  ..-r^  ar^,  a^^^^ 
&c.  are  scies  in  geometrical  progression. 

The  most  usei'ul  properties  of  geometrical  proportion 
and  progression  are  contained  in  the  following  theorems  : 

I.  If  three  quantities  be  in  geometrical  pr<jportion,  the 
product  of  the  two  extremes  will  be  equal  to  the  square 
of  the  mean. 

Thus  if  the  proportionals  be  2,  4,  8,  or  a,  6,  c,  then 
will  2X8=4^,  anda>    =6-. 

>.  Hence,  a  geometrical  mean  proportional,  between 
any  two  quantities,  is  equal  to  the  square  root  of  th§ir 
product. 


AND  PROGRESSION.  96 

Thus,  a  geometric  mean  between  4  and  9  is  =^36  ~-6. 
And  a  geom^^tric  mean  between  a  end  b  is  =^o6*. 

3.  If  four  quantities  be  m  geometrical  proportion,  the 
product  of  the  two  extremes  will  be  equal  to  that  of  the 
means. 

Thus,  if  the  proportionals  be  2,  4,  6,  I  ?,  or  a,  6  c  d- 
then  will  !ii  XI  2=4X6,  and  a  Xrf=6Xc.  '      ' 

4.  Hence,  the  product  of  the  means  of  four  propor- 
tional quantities,  divided  by  either  of  the  extremes,  will 
give  the  other  extreme  ;  and  the  product  of  the  extremes 
div^ed  by  either  of  the  means,  will  give  the  other  mean.    ' 

Thus,  if  the  proportionals  be  3,  9,  5,  1 5,  or  a,  6,  c,  f/; 

Ihen  will  ^=,5,  and  t^^-^9  :    also,  '-2^^,,   and 

aXU     ^ 
=  6, 


c 


5.  Also  if  any  two  products  he  equal  to  each  other, 
either  ot  the  terms  of  one  of  them,  will  be  to  either  of  the 
terms  of  the  other,  as  the  remaming  term  of  the  last  is  to 
the  remaining  term  of  the  first. 

^   Thus,  if  ari=bc,   or  2X  i5=6X.\  then  will  any  of  the 
lollowing  forms  of  these  quantities  be  proportional: 
Directly,  a:  6  : :  c :  r/,  or  :' :  6  •  •   5  :  15 
Invertedly  b:  ay,  d:c,or6:  2  y,   15 :   5. 
Alternately,  a  :  c  y,  6  :  t/,  or  2  :  6  : :  6  :  15. 
Conjunctly,  a  ;  a  +  6  y,  c  :  c  +  r/,  or  2  :  8  : :  5  :  20. 


auLVrv^n'^^'h  ^^°'"'"'*'!'    ">eans  between  any   two   quantities  be  r. 
quired,  they  nia>  be  expre-bstd  as  below  ; 

j/o^^i  a;.cl  l/ab^^  .-.,    two  geometrical  means  betw.en  aand  6. 

-s/a   6,Va^  6^  and  t/«6  =  three  ^■.  om.trical  means  between  c  and  6. 
And  generally, 


(a  6)  .(a         6.)-^,(„         63)"+    ^  any  number  (n)  of  geome- 

trical means  between  a  and  6.    Where  ( j)«^+ 1  is  the  ratio  :  so  that  if  a 

be  multiplied  by  this  it  will  give  the  first'of  these  means  ;  and  this  ]ast  be- 
Tig  again  multiplied  by  the  same,  will  give  the  second  ;  and  so  on 


9«  GEOMETRICAL  PROPORTION 

Distinctly,  a  :  b^a  :  I  c  :  d^c,  or  2  :  4  : :  5  :    10. 
Mixedly,  b-\-a  :  b^a  ::  tZ-f  c  :  dH,  or  8  :  4  :  :  20 :  10. 

In  all  of  which  cases,  the  product  of  the  two  extreme.* 
is  equal  to  that  of  the  two  means. 

6.  In  any  continued  geometrical  series,  the  product  of 
the  two  extremes  is  equal  to  the  product  of  any  two  means 
that  are  equally  distant  from  them  ;  or  to  the  square  of 
the  mean,  when  the  number  of  terms  is  odd. 

Thus,  if  the  series  be  2.  4,  8,  16,  32  ;  then  will 
2X:^2  =  4Xlb=8' 

7.  In  any  geometrical  series,  the  last  term  is  equal  to 
the  product  arising  from  multiplying  the  first  term  by 
such  a  power  of  the  ratio  as.  is  denoted  by  the  number  of 
terms  less  one. 

Thus,  in  the  series  2,  6,  18,54,  162,  we  shall  have 
2X3^=2X*»l=l6x.'. 

And  in  the  series  o,  a?-,  ar^^  ar^,  ar*,  &c.  contmued  to 
terms,  the  last  term  wili  be 

8.  The  sum  of  any  series  of  quantities  in  geometrical 
progression,  either  mcreasmg  or  decreasmg,  is  found  by 
multiplying  the  last  term  by  the  ratio,  and  then  dividing 
the  difference  of  this  product  and  the  first  term  by  the 
difference  between  the  ratio  and  unity. 

Thus,  in    the  series  2,  4,  8,  ib,  32,  64,  128,  i.'o6,  512, 

512X2  —  2 
we  shall  have -——-=1024-2=1022,     the   sum    of 

the  terms. 

Or  the  same  rule,  without  considering  the  last  term, 
may  be  expressed  thus  : 

Find  such  a  power  of  the  ratio  as  is  denoted  by  the 
number  of  terms  of  the  series  ;  then  divide  the  difference 
between  this  power  and  unity,  by  the  difference  between 
the  ratio  and  unity,  and  the  result,  multiplied  by  the  first 
•erm,  will  be  the  sum  of  the  series. 

Thus,  in  the  series  a-{-ar^ar^-\'ar^-\-ar^,  &c.  to  orrK 
we  shall  have 


1^ 


ANi>  PROGRESSION.  97 


Where  it  is  to  be  observed,  that  if  the  ratio,  or  com- 
mon multiplier,  r,  in  this  last  ?enes,  be  a  proper  fraction, 
and  consequent!)'  the  series  a  decreasing  one,  we  shall 
have,  in  that  case, 

a'{-ar-\-ar^-{-ar^-\-ar'^,  &c.  ad  infinitum . 

1  — r 

9.  Three  quantities  are  said  to  be  in  harmonical  pro- 
portion, when  the  first  is  to  the  third,  as  the  difference 
between  the  first  and  second  is  to  the  difference  between 
the  second  and  third. 

Thus,  a,  bj  Cj  are  harnrionically  proportional,  when  a  :  c 
l',a  ~b :  h  —  c,  or  a  :  c  :  :  b  —  a  :  c — 6. 

And  c  is  a  third  harmonical  proportion  to  a  and  6,  when 
_     ab 

10.  Four  quantities  are  in  harmonical  proportion,  when 
the  first  IS  to  the  fourth,  as  the  difference  between  the 
first  and  second  is  to  the  difference  between  the  third  and 
fourth. 

Thus,  «,  6,  c,  d,  are  in  harmonical  proportion,  when 
a  :  d  y,  a  —  b  :  c — c/,  or  «  :  d  y,  b — a  :  d — c.  And  d  is 
a   fourth    harmonical   proportional  to   a,  6,  c,  when  d=t 

-,  in  each  of  which  cases  it  is  obvious,  that  twice  the 

2a — 0 

nrst  term  must  be  greater  than  the  second,  or  otherwise  the 

proportionality  will  not  subsist. 

1 1 .  Any  number  of  quantities,  a,  6,  c,  t/,  e,  &c.  are  m 
harmonical  progression,  if  a :  c  y,  a-^b:  6— c;  6:  d  :: 
b — c  :  c — d;  c  :  e  :  :  c  —  d:  d  —  e\   &c. 

12.  The  reciprocals  of  quantities  in  harmonical  pro- 
gression, are  in  arithmetical  progression. 

Thus,  if  a,  6,  c,  d,  e,  &c.  are  in  harmonical  progres* 

3\oD,  -,  -7»  -,  J,  -,  «c.  Will  be  m  arithmetical  progressioo. 


98        GEOMETRICAL  PROPORTION,  &c. 

13.  An  harmonical  mean  between  any  two  quantities 
is  equal  to  twice  their  product  divided  by  their  sum 

Thus,  — —r  =  an  harmonical  mean  between  a  and  b*. 

EXAMPLES. 

i.  The  first  term  of  a  geographical  series  is  1,  the  ratio 
2,  and  the  number  of  terms  10  ;  what  is  the  sum  of  the 
series. 

Here  1  X20-l  X512=512,  the  last  term. 

.        512X^^—1      1024— I  ^       , 

And  — == =  102'"',  the  sum  required. 

2.  The  fit  St  term  of  a  geometrical  series  is  -,  the  ra 

tio  -,  and  the  number  of  terms  5  ;  required  the  sum  of  tho 
o 

aeries. 

Here  ix  (1)  *=i X^=^4- , he  last  te™. 

And  ^-^_^i=i-|l.=_.  X-=_,  ,hc  sum. 

3.  Required  the  sum  of  1,2,4,  8,  16,  32,  &c.  contt 
nued  to  i'O  terms.  Ans.  1048575. 

4.  Required  the  sum  of  I,  -,  -,  -,  --,  --,  &c.  continu- 

2    4    8    16    tit  jow 

ed  to  8  terms.  Ans.  1  - — . 

128 

6.  Required  the  sum  of  I,  -,  -,  — ,  — ,    &^.    continued 
o    0    27    81  QQ^  J 

to  10  terms.  Ans.  1 —-, 

19683 

6.  A  person  being  asked  to  dispose  of  a  fine  horse,  said 
he  would  sell  him  on  condition  of  having  a  farthing  for 

*  In  addition  to  what  is  here  said,  it  may  be  observed  that  the  ratio  of  tw. 
squares  is  frpquently  called  duplicate  ratio  ;  of  two  square  roots,  snh-dupl: 
cate  ratio  ;  of  two  rnbr  ?,  fripUrnte  ratir  ;  nnd  nf  (wo  ''•ibe  n'0?«,  rub-iripl 
C^te  ratio;  &c 


Of  equations.  99 

the  first  nail  in  his  shoes,  a  half- penny  for  the  second,  a 
penny  for  the  third,  twopence  for  the  fourth,  and  so  on, 
doubling  the  price  of  every  nail,  to  82,  th'^  number  of  nails 
in  his  four  shoes  ;  what  would  the  horse  be  sold  for  at 
that  rate  ?  Ans.  4473924/.  5.«r.  '6^d. 


Of  equations. 

Thl  Doctrink  op  EciUATr>Ns  is  that  branch  of  algebra* 
which  treats  of  the  methods  of  determinin  .  the  values  of 
unkoown  quantities  by  means  of  their  relations  to  others 
which  are  known. 

This  is  done  by  making  certain  algebraic  expressions 
equal  to  each  other  (which  formula,  in  thnt  case,  is  called 
an  equation),  and  then  working  by  the  rules  of  the  art,  till 
the  quantity  S(»ught,  is  found  equal  to  some  given  quantityj 
and  conseq  lently  becomes  known. 

The  terms  of  an  eq  lation  are  the  quantities  of  which 
it  is  co<npo8ed  ;  and  the  parts  that  star^.d  on  the  right  and 
left  of  the  sign  =,  are  called  the  two  members,  or  sides,  of 
the  equation. 

Thus,  if  r=a  -f-6,  the  terms  are  r,  «,  and  b  ;  and  the 
meaning  nf  the  expression  is,  that  some  qua n it y  a;  stand- 
ing on  the  left  hand  side  of  the  equation,  is  equal  to  the 
sum  of  ihe  quanuties  a  and  6  on  the  right  hand  side. 

A  simple  equation  is  that  which  contains  only  the  first 
power  of  the  unknown  quantity  :  as, 

x-\-a~'dh^  or  ar  ~  be,  or  2«  -^-3''~~  5h~  ; 

Where  a-  denotes  the  unknown  quantity,  and  the  other 
letters,  or  numbers,  the  known  quantilies. 

A  compound  equation    is   that  which    contains  two  or 
more  different  powers  of  the  unknown  quantity  :  as, 
x^-^ax=b,  or  r* — 4a;- -|- 3^=25. 

Equations,  are  also  divided  into  different  orders,  or  re- 
ceive particular  names,  according  to  the  highest  power  of 
the  unknown  quantity  contained  in  any  one  of  their  terms  : 
as  quadratic  equations,  cubic  equations,  biquadratic  equa- 
*Jons,  &c. 

Thus,  a  quadratic  equation  is  that  in  which  the  unknown 


100  Of  equations. 

quantity  is  of  two  dimensionsj  or  which  rises  to  the  second 
power  ;  as. 

x'—20  ;  x^-\-(iT=b,  or  3x"--\-l(k!=l00. 

A  cubic  equation  is  that  in  which  the  unknown  quantit} 
is  of  three  dimensions,  or  which  rises  to  the  third  power 
as, 

a:'==27  ;   ^a?''— 3a;- 35  ;  or  a:'' — nx'-hbx  =  c. 

A  biquadratic  equation  is  that  in  which  the  unknown 
quantity  is  of  four  dimensions,  or  which  rises  to  the  fourth 
power:  as   x  =26  ;  Sa?*  —  4x  =  6  ;  or   x' — ax^-\-bx'~ — ca 

=  6/. 

And  so  on  for  equations  of  the  5th,  6th,  and  other  high- 
er orders,  which  are  all  denominated  accordinjr  to  the  high- 
est power  of  the  unknown  quantity  contained  in  any  one 
of  their  terms. 

The  root  of  an  equation  is  such  a  number,  or  quantity^ 
as,  being  substituted  for  the  unknown  quantity,  will  moke 
both  sides  of  the  equation  vanish,  or  become  equal  to  each 
other. 

A  simple  equation  can  have  only  one  root  ;  but  every 
compound  equation  has  as  many  roots  as    it  contains  di 
ijiensions,   or  as  is  denoted    by  the  index   of  the   highest 
power  of  the  unknown  quantity,  in  that  equation. 

Thus,  in  the  quadratic  equation  a;"-l-2a;=15,  the  root, 
or  value  of  x,  is  either  -f  3  or  -  5  ;  and,  in  the  cubic 
equation  x' — 9a;  +2()a;  --V4,  tne  roots  are  2,  3,  and  4,  as 
will  be  found  by  substituting  each  of  these  numbers  for  a:. 

In  an  equation  of  an  odd  number  of  dimensions,  one  of 
its  roots  will  always  be  real  :  whereas  in  an  equation  of 
an  even  number  of  dimen.-ions,  all  its  roots  may  be  imagi- 
nary ;  as  roots  of  this  kind  always  enter  into  an  equation 
by  pairs. 

Such  are  the  equations  a;^  -6a;-{-  14=0,  and  x^ — 2x'^ — 
9a;"-fl0a:4-50=0.* 

*  To  the  properties  of  equations  above-mentioned,  we  may  iiere  farther 
add: 

1.  That  the  sum  of  all  the  roots  of  any  equation  is  equal  to  the  coefficient 
of  the  second  term  ol  that  equation,  with  its  sign  changed. 

2.  The  sum  of  the  products  of  every  two  of  the  roots,  is  eqnal  to  the  cc 
efficient  of  the  third  term-,  without  any  change  in  its  sign. 


SIMPLE  EQUATIONS,  &c.  101 

OF   THE 

RESOLUTION  of  SliMPLE  EQUATIONS, 

Containing  only  one  unknoxen  Quantity. 

The  resolution  of  smple,  as  well  as  of  other  equations, 
is  the  disengaging  the  unknown  quantity,  in  all  such  ex- 
pressions, frotn  the  otaer  quantities  with  which  it  ia  con- 
nected, and  making  it  stano  alone,  on  one  side  of  the  equa- 
tion, so  as  to  be  equal  to  such  a.-j  are  known  on  the  other 
side  ;  for  the  perfortninir  of  which,  several  axioms  and 
processes  are  required,  the  niost  useful  and  necessary  of 
which  are  the  following  ;* 

CASE  I. 

Any  quantity  may  be  transposed  from  one  side  of  an 
tjquation  to  the  other,  by  changing  its  sign  ;  and  the  two 
members,  or  sides,  will  still  be  equal. 

Thus,  if  x-\-'d~l ;  then  will  t=7-3,  or  a  =4. 

And,  if  x-.  4  +  6=8  ;  then  will  r  — 8-f4     6=6. 

Also,  if  X — a+6  — r—rV  :  then  will  x=a  -  b-^-c — d. 

And,  if  4a:- 8  =3x +20;  then4.r-3r=20+8,>nd  con= 
^equently  x=28. 

3.  The  sum  of  the  products  of  every  three  terms  of  the  roots,  is  equal  (c 
.^ft  coefficient  of  tUe  fourth  term,  with  its  sifiii  changed. 

4.  And  soon,  to  the  last,  or  absolute  terra,  which  is  equal  to  the  product  of 
si'l  IfiG  roots,  with  the  sign  chanj^ed  or  not,  according  as  the  equation  is  of 
an  odd  or  an  even  -number  of  dimensions.  See,  for  a  more  particular  ac- 
count of  the  general  theory  of  equation*.  Vol.  II.  of  Bonnycastle's  Treatise 
'>r)  Algtbra,  8vo.  1820;  or  Ryan's  Elementary  Treatise  on  Algebra,  12mo. 
3824.  Ed. 

*  The  operations  required  for  the  purpose  here  mentioned,  are  chiefly  such, 
js  are  derived  from  the  following  simple  and  evident  principles  : 

1.  If  the  same  quantity  be  added  to,  or  subtracted  from,  each  of  two  equal 
quantitie«,  the  results  will  still  be  equal  •,  which  is  the  same,  in  effect,  astak' 
ing  any  quantity  from  one  side  of  an  equation,  and  placing  it  on  the  other 
side,  with  a  contrary  sign. 

2.  If  all  the  terms  of  any  two  equal  quantities,  be  multiplied  or  divided, 
Sy  the  same  quantity,  tlie  products,  orquotienfs  thence  arising,  will  be  equal, 

3.  If  two  quantities,  either  simple  or  compound,  be  equal  to  each  other, 
any  like  powers,  or  roots,  of  them  will  also  be  equal. 

All  of  which  axioms  will  be  found  sufficieniiy  illustrated  by  the  processes 
•arising  out  of  the  several  examples  annexed  to  the  six  different  cases  giver: 

K  2 


i02  SIxMPLE  EQUATIONS. 

From  this  rule  it  also  follows,  that  if  a  quantity  be  founc 
on  each  side  of  an  equation,  with  the  same  sign,  it  may 
be  left  out  of  both  of  thera  ;  and  that  the  signs  of  all  the 
terms  of  any  equation  may  be  changed  from  -f-  to  — ,  or 
from  —  to-f-,  without  altering  its  value. 

Thus,  if  x-\-h  —  7-\-5;  then,  by  cancelHng,  x=7. 

And  if  a^x=b—c  ;  then,  by  changing  the  signs,  a:—. 
<3=c — b,  orx=a-^c — b. 

I-XAMFLKS  FOR  PKAtTlCt. 

1.   Given  2x  +  3^  x-f-lT  to  find  x.     Ans.  a:  =  14. 
"2.  Given  ox — 9- 4x+7  to  find  x.     Ans.  x  =  16. 

3.  Given  x4-9— 2=4  to  find  X.     Ans.  x  =  — 3. 

4.  Given  9.f— 8— 8x-5  to  find  X.     Ans.  x=3. 

5.  Given  7x -1-8-3  =  6x+ 4  to  find  X.     Ans.  x=— K 

a\SE.  II. 
If  the  unknown  quantity,  in  any  equation,  be  niulti{ji: 
ed  by  any  number,  or  quantity,  the  multiplier  may  be  tak- 
en away,  by  dividing  all  tho  rest  of  the  terms  by  it  ;  ant 
if  it  be  divided  by  any  number,  the  divisor  may  be  take' 
away,  by  multiplying  all  the  other  terms  by  it. 

Q 

Thus,  if  ax=Sai~c  ;  then  willx=36 . 

u 

And,  if  2x4-4=16  ;  then  will  x-l-2  =  r, 

orx=^8-^'  =  6. 

Also,  if  ---54-3;  then  will  x  =  10-|-G  =  ltJ. 

2x 
And,  if-— — 2=4;  then    2x--6 -12,   or,    by  division 

o 

^--3=6,  orx=9. 

EXAMPLES  FOR   PRACTICE. 

1.  Given  16X-1-2  =  34  to  finder.     Ans.  x  =  2. 

2.  Given  4r  --8=  -  3x-}-13  to  find  x.     Ans.  x~3. 

3.  Given  lOx— 19=7x  +  17  to  find  r.     Ans.  x=12. 

4.  Given  8.r— 3-{-9=--7x-F94-27  to  find  x. 

Ans.  X— 2 
4d 

5.  Given  2ax—dah=l^LL      Ans.  x=6-| — j-. 

at 


SIMPLE  EQUATIONS.  103 


CASE  III. 

Any  equation  may  be  cleared  of  fractions,  by  multiply 
ing  each  of  its  terras,  successively,   by  the  denominators 
of  those  fractions,  or  by  multiplying  both  sides  by  the  pro- 
duct of  all  the  denominators,  or  by   any  quantity  that  is  a 
multiple  of  them. 

Thus,  if--| — =5,  then,  multiplying  by  3,  we  have  a;-f 
——15;  and   this,  multiplied  by  4,  gives  4.x-\-3x=Q0  : 

ivhence,  by  addition,  7x=60,  or  x=-—=8~. 

'    '  7        7 

And,  if--l — =10  ;  then,  multiplying  by  12,  (which  is  u 
multiple  of  4  and  6,)  3.i-i-2x=l20,  or  5x— 120,  or  rr==: 

5 

It  also  appears,  from  this  rule,  that  if  the  same  number, 
wr  quantity,  be  found  in  each  of  the  terms  of  an  equation, 
either  as  a  multiplitr  or  divisor,  it  may  be  expunged  from 
all  of  them,  without  altering  the  result. 

Thus,  if  ax=ab-i-ac  ;  then  by  cancelling,  x=h-\-c, 

X      0       c 

\ud,  if  -i —  =  -;  then  x+b=c,  or  x^c  —  h. 
a     a     a 


EXAMPLES  FOR  PRACTICE. 


3a:     X 
1.  Given  —=-+24  to  find  i-.  Ans.  x— 19^. 

%  Given  ^+^4-^^=62  to  find  x.  Ans.  a;=60, 

o     5     2 

1    Given  ir5+^=20-^-  to  find  x. 

Ans,  x=9 


i04  SIMPLE  EQUATIONS. 

4.  Given— jr — j — ^—=16 to  find  .r. 

D.  Given—; — H — = to  find  x. 

b        c      a         d 


Ans.  x=J3 


acd-\-abd — 2chd 
CASE  IV. 

If  the  unknown  quantity,  in  any  equation,  be  in  the 
form  of  a  surd,  transp.we  the  terms  .so  that  this  may  stand 
alone,  on  one  side  of  the  pquatu»n,  and  the  remaining 
terms  on  the  other  (by  <'a-e  I)  ;  then  involve  each  of  the 
sides  to  such  a  power  as  curresponds  with  the  index  of 
the  surd,  and  the  equation  will  be  rendered  free  from  any 
irrational  expression. 

Thus,  if  y/a;  — 2  — 3  ;  then  will  ^a'=3H-2'~5,  or,  by 
squaring,  a:=5^=25. 

And  if  v'(3x4-4)  =  5  ;  then  will  3x-f  4=25,  or  3x  =  25 

21 

-4=21,  ora:=--  =  7. 
o 

Also,  if  i^(2x-l-3)4-l=8  ;  then  will  :y(2a-f3)=S~4 

=  4,  or  2x-r3=4=64,  and  consequently  i'a~  64-3  =  6 

61      „    1 

.■■  .=  2—30-. 

EXAMPLES  FOR  PRACTICE. 

1.  Given  2v/x  + 3=9  to  find  a*.  Ans.  x=9. 

2.  Given  v'(x+l)-2=3to  find  a:.  Ans.  x=24. 

3.  Given  3/(3a:4-4)+3=t)  to  find  x.  Ans.  a;=7|. 
•I.  Given  v/(4-fa;)=4--^a:  to  find  x.  Ans.  a;=2f 
'>,  Given  y{^w'-\-x')  =  ^^{Ah'-\-x^)  to  find  x, 

Ans..T=^(-2^-). 


SIMPLE  EQUATIONS  105 

CASE  V. 

if  that  side  of  the  equation  which  contains  the  unknown 
quantity,  be  a  complete  poner,  the  equation  may  be  re- 
duced to  a  lower  dimension,  by  extractinij  the  root  of  the 
said  power  on  both  sides  of  the  equation. 

Thus,  if  rc^=8l;  then  3:  =  ^8>=9;  and  if  a;==27, 
then  a- =  ^27=3. 

Also,   if  3x-- 9=24;    then  3  r =24-1-9=3.?,  or    x^=^ 

33 

—-=1 1,  and  consequently  x=^ll. 

And,  if  x^-f  6.r  +  9  =  27  ;  then,  since  the  left  hand  side 
of  the  equation  is  a  complete  square,  we  shall  have,  by 
extracting  the  roots,  a:+3=^27=v^(9X3)=3v^3,  or  or 
=3v/3— 3. 

EXAMPLES  FOR  PRACTICE. 

1 .  Given  Qa;^— 6=30  to  find  x.  Ans.  x=2. 

2.  Given  a-f  9=3h  to  find  x.  Ans.  x=3. 

8  1 

3.  Given  x^-\-x-{-l= —  to  find  x,  Ans.  x  — 4. 

4.  Given  x^=ax-j— — =6"  to  find  x.  Ans.  a:=6-^ 

4  •  * 

5.  Given  x--{-14x-r49-  >2l  to  find  a:.       Ans.  .'r=4. 

CASE.  IV. 

Any  analogy,  or  proportion,  may  be  converted  into  an 
equation,  by  making  the  product  of  the  two  extreme  terms 
equal  to  that  of  the  two  means. 

Thus,  if  3x:    16  ::  5:  b;  then  3xX6=16X5,  or  IBx 

80     40        4 
-80,orx=-=-=4-. 

2x  2cx 

And  if  — -  :  /I  : :  6  :    c  ;   then  will  —^^ab,  or  2cx~3a6  ; 

o  O 


106  SIMPLE  EQUATIONS. 

or,  by  division,  x=^— . 

Also,  if  12— X  :   ;J : :  4  :  1  ;  then  U-x==-^=2x,  or  2a 

12 

f  ^=12,  and  consequently  x=—=4. 


EXAMPLES  FOR  PRACTICE. 


.  Given-a::a::  56c:  Cf/ to  find  x.       Ans.  a;=     ■— = 
4  3(i 


2 

2.  Given  10-a^  :  ^  x  :  :  3  :  I  to  find  a:.  Ans.  a:=3f 

3.  Given  S  +  ^x  :  4x  :  :  8  :  2  to  find  x,  Ans.  a;  =  l. 

4.  Given  x  :  6  — a;  :  :  2  :   4  to  find  x,  Ans.  x=2. 

a2 

5.  Given  4x  :  m  : :  9^x  :  9  to  find  r.  Ans.  x=-— . 

16 


BllSCELLANEOUS  EXAMPLES. 

1.  Given  ox  -  I5— 2x-{-6  to  find  the  value  of  x. 

Here  5x-.2t^6+15,  or3x=6-fl5-  21;  and   there- 

21     ^ 
lore  x=— =7. 
t> 

2.  Given 40..  t;r  -  16=  120— l4.r.  to  find  the  value  of  x. 
Here  14x— o*  =  120  -  40+ 16  ;  or  8x=l3b— 40=96  ; 

9  . 
and  therefore  t  =  -  =12. 

8 

3.  Given  3x^       0r  =  8r-l-x^,  to  find  the  value  of  x. 
Here  3jr — 10=  -  ^x,  by  dividing   by  x  ;  or  3x  — x=8-f 

10=18,  by  transp->--.iion. 

1 8 
And  consequenily  '2r  =  18,  or  x  ---=9. 

4.  Given  67x3-12a6x2-2ax^+6ax^  to  find  the  value 
of  X. 


SIMPLE  EQUATIONS.  i07 

Here  2j:— 4^y=x-|-2,  by  dividing  by  3ai2  ;  or  2x — x— 
iJ+^6  ;  and  therefore  x=4b-\-2. 

5.  Given  x^  -I- 2x+  1  =  1 6,  to  find  the  value  of  x. 
Here  a:-h'=4,  by  extracting  the  square  root  of  each 

iide. 
And  therefore,  by  transposition,  x=4—  1  —  3, 

6.  GiveH  bax  -36=i?dx+c,  to  find  the  vahie  of  x. 
Here   5aj  -  2U'= c-{-^b;    or  (5a  — 2c^)x=c-|-36  ;   and 

herefore,  by  division,  x= ,. 

XXX 

7.  Given -f-'-=10,  to  find  the  value  of  x. 

2  3     4 

2r     2x                                      6x 
Here  x -H =20  ;  and  3x  — 2xH =G0  ;    or  I2a. 

3  4  4 

~-8x+6x=240  ;  whence  1  Ox =240,  or  x=^ 24. 

X  —  3     X              X  —  19 
3.  Given—- — (-5=20 — -,  to  find  the  value  of  x 

Here  x-S-f  ^=40-x-|-l9;  or  3x-9-f-2x=  120---3x 

4-57;  whence  3x-f-2x+3x=120-f67+9 ;  that  is  8x= 

186,  orx=23i. 

2x 
9.  Given  -y/— -[-5  =  7,  to  find  the  value  of  x. 

2x  2x 

Here  v/-^  =  7~5— 2:  whence,  by  squaring,  —=22=4; 

O  o 

and  2x=12,  or  x=6. 

2a2 
)  0.  Given  x-f- 1/  (n^  -f-a-^  "l  = — t-^, — o — »  to  find  the  va- 

lue  of  X. 

Here  x^{a^-{-x^)-{-a^'{'X^=2a^;  or  x-v/  (a^-j-x^')  =:. 
d^— X-,  and  x^  (a^+x^^)  =  a^-2alr2-f-x^;  whence  aV 
-l-x^  -_-  a*-.2aV4-x^    and   fi-x^  =  a^~2ay ;    therefore 

3a-x-=a%  orx  =  — -  =  — ;    and  consequently  x=v'~ 

ofl"  o  *-     , 

:=ia  yj  o =" v^Q  ~  oa/^  '  *^^^  answer  required- 


OS  SIMPLE  EQUATIONS. 


EXAMPLES  FOR  PRACTICE. 

1.  Given  3x— 2-^24=31,  to  find  the  value  of  x. 

Ans.  x=3- 

2.  Given  -i  —  Sy  =14 — 1  It/,  to  find  the  value  oi'y. 

Ans.  y=5. 

3.  Given  a-|-l8=3ar  —  5,  to  find  the  value  ofx. 

Ans.  i=Ilf 

X       X 

4.  Given  a;-f--f--=n,  to  determine  the  value  of  :r. 

'<£      o 

Ans.  1  =  6. 

X 

5.  Given  2x — --f-l—5x  — 2,  to  find  the  value  of  a:. 

lit 

Ans.  x=--. 
7 

6.  Given  jr+o — r=—,  to  determine  the  value  of  rr. 

2      o      4      10 

Ans.  a:=l-. 
5 

7.  Given  — jr — |-5='l >  to  find  the  value  of  .r. 

<it         o  4 

Ans.  a;=3—- . 

JO 

9.  Given  2-f-v/3a:=.y4-f  5x,  to  find  the  value  of  rr. 

Ans.  .r=12. 

3.  Given  x-f-a= — ; — ,  to  find  the  value  of  x. 

a-f-x 

Ans.  x=— -. 

.  2a 

iO.  Given  v^x4-v/a+x=   ..^^^x    to  find  the  value 

•f  X.  Ans.  x=x. 

o 

^,     _.        ax— 6  ,  a     bx     bx — a  ^    ,     ,  , 

11.  Given — ho=-s 5 — j    ^o   find  ^^e   value 

4  o       ^  o 


SIMPLE  EQUATIONS,  109 

oi  5:.  Ans.  x=~ -r- 

3a  — 26 

12.  Given  ^a''+x^={/h'+x\  to  find  the  value  of  x. 

Ans.  x—^   ^  „-, 


13.  Given -v/a4-x-|-y^a-a:=-v/aa',   to  find   the  value 
of  x.  Ans.  a;=-2-r— . 

14.  Given h — ^==^j  to  determine  the  value  of  a'. 

1-1- a;     l^x 

Ans.  X=:y^ r . 

15.  Given  a-f-x=v'ftM^xv^(6M^)j  to  find   the  value 

t)f  a:.  Ans.  a;=-- a. 

4a 

16.  Given  iv^(a;2-|,3a2)—L^(a;2-.3a2)=a:v^a,   to  find 

9a^ 
iho  value  of  a:.  Ans.  x=i/  ^ 

^  4— 4a 

17.  Given  v^(a-ra')-fv^(«— x)=t,   to  find  the  value 
of  rr.  kns.x=ty{'ia-h% 

IS.  Given  V(a+x)-t-^(a  -x)— &,  to  find  the  yalue  of  x. 


/   o     /6^-2a\^ 
Ans.a:=Va^-^-^-^' 


19.  Given  ^a-\'^x=^y/ax,  to  find  the  value  of  x, 

Ans.  x=- 


20.  Given  v/f^^  j+^/^-^l=a,    to  determine 
?he  value  of  x.  Ans.  a;= — » 


21.  Given  v'(a2-i-aa;)=a—y^(a2-.rta;),  to  find  the  va« 
lue  of  X.  Ans.  x=-^3, 


no  SIMPLE  EQUATIONS, 

22.  Given   ^/(,a'  —  x')-{-xy/{a'^\)=a''^{l-^x'),    ic 
find  the  value  of  a;.  Ans.  x=./f^-^l\ 

23.  Given -/(a;+«)=c-  -/(x-l-^),  to  find  the  vakie  of  a:. 

24.  Given  ^-L.+y'-i-  =V-i^,,  to  find  the  va- 


lue  of  a;.  Ans. 


64-C' 


Of  the  resolution  of  simple  equations,  containing  izn-i 
unknown  quantities. 


When  there  are  two  unknown  quantities,  and  two  inde- 
pendent simple  equations  involving  them,  they  may  be  re- 
duced to  one,  by  any  of  the  three  following  rules  : 


.  RULE  I, 

Observe  which  of  the  unknown  quantities  is  the  least 
involved,  and  find  its  value  in  each  of  the  equations,  by 
the  methods  already  explained ;  then  let  the  two  values, 
thus  found,  be  put  equal  to  each  other,  and  there  will  arise 
a  new  equation  with  only  one  unknown  quantity  in  it,  the 
value  of  which  may  be  found  as  before*. 


*  This  rule  depends  upon  the  well  known  axiom,  that  things  which  arf 
equal  t©  the  same  thing,  are  equal  to  each  other  ;  and  the  two  followiotr  me- 
Uicras  are  founded  on  principles  which  are  equally  simple  and  obviou- 


SIMPLE  EQUATIONS.  Ill 


EXAMPLES. 


1 .  Oiven  j  5^_2v—  10  i  ^^  ^"^  '^^®  values  of  x  and  y. 
Here,  from  the  first  equation,  x=~ -, 

And  from  the  second,  a;= 

5 

vYhence  we  have =-= ^- 

^  o 

Orll5— 15i/=20+4y,  or  1%=  115-20==  95. 
That  is,  2/=— =5,  and  x=^^~- — =4, 

2.  Given  )    3^^a(   to  find  the  values  of  o;  and  ?/. 

Here,  from  the  first  equation,  x^a  —  y. 
And  from  the  second  rr=6-f-?/, 
Whence  x  -^y^b-^-y^  or  2y=^a  —  by 

And  therefore  2/=-t~;  ^^^  a;=ft— ?/. 

^     ,         ...  a  — 6     0  +  ^ 

Or,  by  substitution,  a*=a ^ . 

■^  '  2  2 

3.  Given   \  tltt^'^o  ^   to  find  the  values  of  x  and  y. 
Here,  from  the  first  equation,  a;=14 — ^, 

o 

And  from  the  second,  a;  =2 4 --^ 

2 

Therefore,  by  equality,  14  — -^=24 -^ 

And  consequently  42  — 2;/^72 — ^, 


112  SIMPLE  EQUATIONS. 

Or  by  multiplication  84  —  4?/=  1 44  —  9)/ ; 
And,  therefore,  also  52/=144— 84=60, 

Or,  by  division,  .t=-~-=12,  and  a;=14 — 3-=^. 


EXAMPLES  FOR  PRACTICE. 

1.  Given  4x-l-i/=34,  and  4^4--'*''—  16,  to  find  the  values 
of  x  and  y.  Ans.  a'=S,  y=2, 

2.  Given  2x'^3y=ie,  and  3x-2?/=ll,  to  find  ttie  va- 
lues of  X  and  y.  Ans.  a-— 5,  2/  =  2. 

3.Given|+f=l,and|+|=^,   tofindthc 

values  of  a;  and  y.  Ans.  a=4,  2/=-}' 

4.  Given   )  1^^  J^~t  ?  to  find  a;  and  y. 

Ans.  rc=a-fZ?,  and2/=ia— |c). 

f  ^  >  y     "s 

\  2*^3        f 

5.  Given  ^  ^     ^        >  to  find  x  and  t/. 

3"~2~'    }  Ans.  x'~12,  andsr=6. 


6.  Given  /  2~^3  >  to  find  rr  and  y, 

I  a: :  2/  : :  4  :  3  ?  -^"s.  a;  =  12,  and  y"-9.. 

2a.'     3u 

7.  Given  a-4-V=S0.  and  — ==-^,  to  find  x  and  ?/. 

Ans.  a;=42-A.,  and  1/— 37j-^. 

X 

3.  Given  7^  —  6=--,  and  x—y+Q,  to  find  a;  and  y. 

Ans.  x=24,  and  2/=I& 

RULE  II. 

Find  the  values  of  either  of  the  unknown  quantities  in 
that  equation  in  which  it  is  the  least  involved  ;  then  sub- 
stitute this   value  in  the  place   of  its  equal  in    the   other 


SIMPLE  EQUATIONS.  US 

equation,  and  there  will  arise  a  new  equation  with  only 
one  unknown  quantity  in  it  ;  the  value  of  which  may  be 
found  as  before. 

EXAMPLES. 

1.  Given   ]  o    _     =2    i  ^°  ^"^  ^^^^  values  of  x  and  p. 
From  the  first  equation,  x*— 17— 22/  ;  which  value,  be- 
ing substituted  for  x,  in  the  second, 

gives  3{\7-.2y)^y  =  2. 

Or  51-62/— 2/=2,  or  7y=51— 2=^49, 

49 
Whence  «/=— =7,  and  a;=l7  — 2?/=3. 

2.  Given   <  rr'Z.   =:  3i^^  ^"^  ^^^®  values  of  x  and  y. 
From  the   first  equation,  x=13 — y  ;  which  value  being 

substituted  for  a:,  in  the  second, 

Gives  13«7/-?/  =  3,  or  2^=^13-3  =  10, 

Whence  2/=^ =5,  and  a;=13  "-^— 8. 

3..  Given  <     '  "'gV.  2_1.    {  to  find  the  values  of  a-  and  y. 
Here  the  analogy  in  the  first,  turned  into  an  equation, 

gives  ox'^ay,  or  ^  —  -ri 
And  this  value,  substituted  for  x  in  the  second, 

gives  [~y-\-7f=C,  OY'^^+y=C, 

Whence  we  have  ttV+^V=^^^3  or  y^== p-j— 

c  c 

And,  consequently,  y^^V-^^^-^,  and  x=a^-^j--^^. 

EXAMPLES  FOR  PRACTICS;. 
X  11 

1.  Given  -+7i/=99,  and  |+'7^=51,  to  find  the  values 

of  0:  and  y,  Ans.  a:=7,  and  y=M. 

l2 


■14  SIMPLE  EQUATIONS. 

2.  Given  I- 12=1+8,  ,.d^+?-8=!l^V2r, 

to  find  the  values  of  x  and  y.  Ans.  a:— 60,  ?/=40. 

3.  Given  a;-{-2/=«,  and  :e^ — 2/"—^)  ^^  fi"^  ^^^6  values  ot 

X'  and  y.  Ans.  ^=-27-'  2/=-^7"- 

4.  GivenSa:— 3?/=l50,  and  10a;-l-l52/=825,  to  find  a 
and  y.  Ans.  a=45,  and  i/=25. 

5.  Given  a;+3/=16,  and  .t  :  1/  : :  3  :  I,  to  find  x  and  y. 

Ans.  a;=  12.  and  v/=4. 

0.  Given  :e-r|=12,  and  y+^  — ^>  to  find  a;  and  y. 

x\ns.  rr=  1 0,  and  ?/=4. 
?.  Given  x*  :  t/  ::  3:2,  and  a;^— 3/-=20,  to  find  x  and 2/. 

Ans.  a:=6,  and  2/=4. 

8.  Given '^-12-^+13  and  ^+^+16  =  ?^  + 
2  4  5        3  4 

i>7,  to  find  x  and  ?/■.  Ans.  .t  =  60,  and  t,'~20. 

RULE  III. 

Let  one  or  both  of  the  given  equations  be  multipHed,  01 
divided,  by  such  numbers,  or  quantities,  as  will  make  the 
term  that  contains  one  of  the  unknown  quantities  the  same 
m  each  of  them  ;  then,  by  addino;,  or  subtracting,  the  two 
equations  thus  obtained,  as  the  case  may  require,  there 
vvili  arise  a  new  equation,  with  only  one  unknown  quantit\ 
m  it,  v/hich  may  be  resolved  as  betlore"^. 


*  The  values  of  the  unknown  quantities  in  (he  t«'o  literal  equations  aar-f 
hy  =  c,  anda'x-|-  h'y  =  c',  may  be  found  in  general  terms,  by  multiplying  the 
irst  by  a',  and  the  second  by  a,  and  then  working  accordii»g  to  the  last  rule, 

,  .-,...,  «c' — ca'         ,         ch'—hd 

when  the  results,  so  determined,  will  be  y=  — - — -— ,andx=3-— — r- 

06  — 6a'  ab — ha,' 

vbicb  solution  may  be  applied  to  any  particular  case  of  this  kind,  by  substi 
rating  the  numeral  of  a,  b,  a',  b\  in  the  place  of  the  letters,  and  observing^ 
v/beu  either  of  thera  is  negative,  to  change  the  signs  accoidingly. 

Where  the  numerator  is  the  diflerence  of  the  products  of  the  opposite  co 
efficients  in  the  order  in  which  y  is  not  found,  and  the  denominator  is  the  dif 
ference  of  the  products  of  the  opposite  coefficients  taken  from  the  orders  tha- 
involve  the  two  unknown  quantities.  Coefficients  are  of  the  same  order  whici" 


SIMPLE  EQUATIONS,  115 

EXAMPLES. 

i.  Given  <    ''^T  >^~"i  ,  i  to  find  the  values  of  a*  and  y. 

First,  multiply   the   second  equation   by   3,   and  it  will 
give  3x-\-6y—42. 

Then,   subtract  the  first   equation  fi-om  this  and  it  will 
give  67/  — 5?/ =4  s;  — 40,  or  y=2. 

Whence,  also,  x=14  — %— 14— 4=10. 

2.   Given  ^  f'^'T?-'^?^  ^   to  find  the  values  of  a:  and  v. 

Multiply  the   first  equation  by  2,  and  the  second  by  5  ; 
then  10a; — by=]ii,  and  !  Ox -1-25?/=: SO. 

And  if  the  former  of  these  be  subtracted  from  the  latter, 

there  will  arise  31?/= 62,  or  i/= — =2. 

Whence,  by  the  first  equation,  x— — — ^u=-— =3. 

o  o 

EXAMPLES  FOR  PRACTICE. 

1.  Given  ^i^+6y=2l,  and '^-=23-5a',   to  find  ;i; 

4  ^  3 

iind  y.  Ans.  :c  =  4,  and  y=-3^ 

2' 

2.  Given  3a;-f  7y— 79,  and  2^~9H-- ,  to  find  x  and  y, 

Ans.  rr  =  10,  and  y=^7. 

3.  Given  30j;-f  40?/=270,  and  50.r+30^=340,  to  find  a: 
and  y.  Ans.  x  —5,  and  ?/=3. 

4.  Given3a~3?/=2x-f2i/  and  x-\-y  •  xy  : :  3  :  5,  to  find 
V  and  y.  Ans.  x^\0,  and  ^=2. 

5.  Given  a:2.?^+a;?/^=30,    and   x^-\-y^=S5,     to    find  .r 
and  y.  Ans.  x=3,  and  ?/=2. 


either  affect  no  unknown  quantity,  as  c  and  C  ;  or  (he  same  unknown  quanti- 
ty in  the  difterent  equations,  as  a  and  a'.  Coefficients  are  opposite  when  they 
aflect  the  diflerent  unknown  quantities  in  the  difierent  equations,  as  a  and^-, 
&■  dnd  b.  'Ed. 


116  SIMPLE  EQUATIONS. 

6.  Given  ~-^-^=—^ 3,  and  8 ^--=2+1'^^- 

find  X  aad  y.  ^   Ans.  x=l2,  and  2/=6. 

7.  Given   x-{-y-  «:  :  a;  —  ?/:  6,   and   x^'-if=c,  to  find 
the  values  of  x  and  y. 

ft-f-^    ,c  a  —  b      c 

8.  Given  ax-\-by=c,   and  i/:c4-«^=/)  to  find  the  values 

of  X  and  y. 

ce  —  6/"  a/"—  dc 

Ans.  a= ±  y=-^-^ 

ae — bd         ae  —  bd 

9.  Given   x-\-y=a,  and  x"—y^=^b,  to  find  the  values  of 

a:  and  2/.  ^"^-'^■=~2^-' ^="2^-' 

10.  Given  x^-\-xy=^aj  and  ^y'+sr^/— ^j  ^°  ^"^  *^®  values 

a  ^ 

of  a'  and  y.  Ans.  a;= — -—         ,  y 


Of  iJie  resolution   of  simple  equations,  containing  three  or 
more  nnknonn  quantities. 

When  there  are  three  unknown  quantities,  and  three 
independent  simple  equations  containing  them,  they  may 
he  reduced  to  one  by  the  following  method*. 

RULE. 

Find  the  values  of  one  of  the  unknown  quantities  in 
each  of  the  three  given  equations,  as  i.f  all  the  rest  were 


*  The  necessity  for  observing  that  the  given  equations  in  this  and  other 
similar  cases  are  so  proposed  as  to  be  independent  of  each  other,  will  be 
obvious  from  the  foUowina;  example  : 

x—2}/-\-  z=.5  \  2x4-y—z  =x  7  ;  a  4-  3]/— 22c=  2  ; 
where,  if  it  were  required  to  deteimine  the  values  of  x,  y,  andz,  it  will  be 
iound  by  eliminating  a:  from  each  of  them,  and  then  equating  the  results,  that 

5?/— 35r  =— 3,  and  53/— 3z  =— 3  ; 
which  equations,  being  identical,  or  both  the  same,  furnish  no  determinate 
answer.     And,  in  effect,  if  the  three  equations  be  properly  examined,  it  wilt 
be  found,  that  the  third  is  merely  the  difference  of  the  first  and  second,  and 
Coiisf:quentlv  involves  no  condition  but  what  is  contained  in  the  other  tWQ. 


SIMPLE  EQUATIONS.  117 

known  ;  then  put  the  first  of  these  values  equal  to  the  se- 
cond, and  either  the  first  or  second  equal  to  the  third,  and 
there  will  arise  two  new  equations  with  only  two  unknown 
quantities  in  them,  the  values  of  which  may  be  found  as 
in  the  former  case  ;  and  thence  the  value  of  third. 

Or,  multiply  each  of  the  equations  by  such  numbers,  or 
quantities,  as  will  make  one  of  their  terms  the  same  in 
them  all  ;  then  having  subtracted  any  two  of  these  result- 
ing equations  from  the  third,  or  added  them  together,  as 
the  case  may  require,  there  will  remain  only  two  equa» 
tions,  which  may  be  resolved  by  the  former  rules. 

And  in  nearly  the  same  way  may  four,  five,  &c.  un- 
known quantities  be  exterminated  from  the  same  num- 
ber of  independent  simple  equations  ;  but,  in  cases  of  this 
kind,  there  are  frequently  shorter  and  more  commodious 
methods  of  operation,  which  can  only  be  learnt  from  prac- 
■  ice*. 


EXAMPLES. 


Given  /    x-{'^^y-{-3z=62\ 


to  find  X,  y,  and  0. 


^'   The  values  of  the  unknown  quantities  in  the  three  literal  equations 

fia;~\-byJ^cz  =>d  ;  a'x~\-lj'i/-*'c'z=  d'  ;  a''x  '^b"'y-\-c''Z'=  d"  ; 

j»iay  be  exhibited  in  jreneral  terms,  like  those  before  mentioned,  as  follows 

db'c" — dc'b''-\-cd'b'' — bd'c''-\-bc'd'' — cbd" 

ab'C' — ac'b"-^ca'b" — ba'c"-\^bc'a" — cb'a" 

ad'c" — nc'd"-\-ca'd'' — da'C'-^dc'a'' — cd'a'' 

ab'c" — ac'b"  -^-ca'b'' — ba  c" •\- be' a" — cb'a', 
abd" —  ad'b''  -i-  da'b" —  ba'dJ'-^-hd'a" — db'a" 

ah'c'' — ac'b" -[-cab" — ba'c"  -i-  hc'a" — cb'a'' 
,..:jjii  formulaj,  by  substitution,  may  be  employed  for  the  resolution  of  any 
iiL'mcral  case  of  this  kiiid,  as  in  the  instance  of  two  equations  before  given, 
The  numerator  of  any  of  these  equations  such  as  z,  a>nsists  of  all  the  dif- 
ferent products,  wh)«ij  can  hf  made  of  three  opposite  coefficients  taken  from 
the  orders  ill  wliich  z  is  not  found;  and  the  dcnofuiaalor  consists  of  all  the 
products  thnt  can  be  made  of  the  three  opposite  coefficisnts  taken  f|om  the 
orders  n-hicU  involve  the  thi'ec  unknown  quantities. 


11$  SIMPLE  EQUATIONS. 

Here,  from  the  first  equation,  a;=29  — ?/ — z. 

From  the  second,  x=62  — 23/  — 32. 
2        ] 
And  from  the  third,  ar=20 — y — z, 

Whence       29 — y— -2=62  —  21/  — Sz^ 

2       1 
And,  also,    ^9^y-z=20^-y^-z, 

From  the  first  of  which  2/=33--2r, 

And  from  the  second,       y=^27—-z, 

Therefore  33 -2^=27-.  ^^,  or  ^=12, 

Whence,  also,  ?/  =  33  -2^=9 

And  a:=29~2/-'=^  =  8. 
^2:r  +  4?/-3^=22y 

2.  Given   (  4z— 2?/  +  62'=  18  ^   to  find  a:,  ?/,  and  r, 

(6x4-7?/— 2r   =63  ) 
Here  multiplying  the  first  equation  by  6,  the   second  by 
3,  and  the  third  by  2,  we  shall  have 

l2a;+24?/— 182=132, 
]2x-   62/-f-15r=54, 
12.T+14</-   2^  =  126. 
And,  subtracting  the  second  of  these  equations  succes 
sively  from  the  first  and  third,  there  will  arise 
30«/— 332=78, 
20t/- 172=72. 
Or,  by  dividing   the  first  of  these  two  equations  by   8. 
and  then  multiplying  the  result  by  2, 
2O1/ -222=52, 
2U2/      172=72. 
Whence,  by  subtracting  the  former  of  these  from  the 
latter,  we  have  52=20,  or  2=4, 

And,  consequently,  by  substitution  and  reduction, 
2/=7,  and  a;  =  3. 

3.  Given  x4-2/+^=53,  a^-}-27/+32=105,  and  x-f  3j/-|- 
•|2=l34j  to  find  the  values  of  x,  y,  and  2. 

Ans.  x=24,  ?/=6,  and2'=23. 


SIMPLE  EQUATIONS.  jl^ 

4.  Given  x-hy^y=Z2,  ^+^=^^15,  and  ^.x'+i 

1 

2/+g^=12,  to  find  the  values  of  a*,  y  and  z. 

Ans.  x'=l2, 2/~20,  .x~30. 

5.  Given  lx+by-h2z=7y,  8a;-{-7i/+92==i22,  and  a;4 
42/4-5^=55,  to  find  the  values  of  a:,  y,  and  z. 

Ans.  ac=4,  y~9,  2=3, 

6.  Given  a;-{-i/=a,  x+z—b,  and  ?/4-2^=c,  to  find  the  va- 
lues of  Xf  y,  and  z. 

Ans.  2/= ^ ,  x= — ^ and  2-=  — -3-, 

33,  to  find  a:,  y,  and  z.       Ans.  a: -24,  y=60,  and  2;=  120. 

8.  Given  z4-2/=x4- 100,  y-2x=2z~  100,  and  z-flOO 
~3x-{-3y,  to  find  or,  y,  and  r. 

on-  ,      ,     Ans.T  =  9V^,2/=45/3-,and;r  =  63,V 

9.  i^ivenx-\-y-{-z=:7,2x^3==y{-3z,  and  5x+5z='Sy 
-1-19,  to  find  Xf  y,  and  z,       Ans.  a;=4,  2/=2,  and  z  =  \. 

10.  Given  32:  +  5i/  -  42'=25,  bx  -  2i/-f -3^=46,  and  3w-f- 
52'-.a;=62,  to  find.r,  ?/,  and  z. 

Ans.  a;=7,  ?/=8,  and  z=d. 

11.  *  Given  a;-hi/+2=13,a-+^+w  =  17,  x+-+w=18, 
and  y-\-z-\-u=2\,  to  find  .1;,  y,  and  u. 

Ans.  a;=2,  t/  -  5,  ^=6,  and  ii=10. 

MISCELLANEOUS  QUESTIONS, 

PRODUCING  SIMPLE  EQUATIONS, 

The  usual  method  of  resolving  algebraical  questions, 
IS  first  to  denote  the  quantities,  that  are  to  be  found,  by  x, 
yi  or  some  of  the  other   final   letters  of   the  alphabet; 

*  This  can  be  resolved  by  proceeding  after  the  same  manner  as  equations 
involving  three  unknov^n  quantities  ;  but  the  resolution  of  it  may  be  greatly 
facilitated,  by  introducing  into  the  calculation,  beside  the  principal  unknowu 
•quantities,  a  new  unknown  quantity  arbitrarily  assumed,  such  as,  for  exam- 
ple, the  sum  of  all  the  rest :  and  when  a  little  practised  io  such  c^Iculatio&s, 
wp.y  become  easy> 


120  SIMPLE  EQUATIONS. 

then,  having  properly  examined  the  state  of  the  questioii 
perform  with  these  letters,  and  the  known  quantities,  by 
means  of  the  common  signs,  the  same  operations  and  rea- 
sonings, that  it  would  be  necessary  to  make  if  the  quanti- 
ties were  known,  and  it  was  required  to  verify  them,  anti 
the  conclusion  will  give  the  result  sought. 

Or,  it  is  generally  best,  when  it  can  be  done,  to  denote 
only  one  of  the  unknown  quantities  by  x  or  ?/,  and  then 
to  determine  the  expression  for  the  others,  from  the  na- 
ture  of  the  question  ;  after  which  the  same  method  of 
reasoning  may  be  followed,  as  above.  And,  in  some  cases, 
the  substituting  for  the  sum  and  differences  of  quantities  ; 
or  availing  ourselves  of  any  other  mode,  that  a  proper 
consideration  of  the  question  may  suggest,  will  greatly  fa- 
cilitate the  solution. 

1.  What  number  is  that  whose  third  part  exceeds  itp 
fourth  part  by  16  1 

Let  3;=the  number  required. 

Then  its  -  part  will  be  -a',  and  its  -  part  -x. 
o  o  4  4 

And  therefore  -x  --3:=16,  by  the  question, 

That  is,  .T--a;=4S,  or  4^:— 3j;=192, 

Hence  a:— 192,  the  number  required. 

2.  It  is  required  to  find  two  numbers  such,  that  thei: 
mm  shall  be  40,  and  their  difference  16. 

Let  X  denote  the  least  of  the  two  numbers  required. 
Then  will  x-\- 1 6=to  the  greater  number. 
And  a;+x -1-1 6=40,  by  the  question, 

That  is,  2.x =40— 16,  or  x=— =  12=least  number. 

And  x-f  16— 12-f  16=28=  the  greater  number  required, 

3.  Divide  1000/.  between  a,  b,  and  c,  so  that  a  shall 
have  72/.  more  than  b,  and  c  100/.  more  than  a. 

Let  a;=E's  share  of  the  given  sum, 
Then  will  a:-f  72=a's  share, 
And  .r+  I72=c's  share- 


SIMPLE  EQUATIONS.  121 

Hence  their  sum  is  x+^r+^a+or-f-ns, 
Or  :^a;+244=  lOOO,  by  the  question. 
That  is,  Sx=^  1000—244=756, 

756 
Or  0;=-— =252/.  =  B.'s  share, 

Hence  a;  +  72=324/.=A.'s  sharCc 
And  X  4-172=  424^'  ==c/s  share. 
Also,  as  above,  252/.=b.'s  share- 


Sum  of  all  =100   \  the  proof. 

4.  It  is  required  to  divide   1000/.  between  two  persoiis, 
:0  that  their  shares  of  it  shall  be  in  the  proportion  of  7  to  9. 

Let  r-=  the  first  person's  share, 

Then  will  1000 — x=  second  person's  share, 

And  X  :  1 000 — x  : :  7  :  9,  by  the  question, 

That  IS  9x'=(i000— a)X7=7000— 7a;, 

7000 
Or9a;+7a;=7000,orx=~-^--=437Z.   IO5.   =  1st  share, 
lo  ' 

and  1000— x=1000-437/.    10s.=  562/.   I05.=2d  share« 

5.  The  pavinor  of  a  square  court  with  stones,  at  25.  a 
yard,  will  cost  as  much  as  the  enclosing  it  with  pallisadeSy 
at  55.  a  yard  ;  required  the  side  of  the  square. 

Let  a:=length  of  the  side  of  the  square  sought, 
Then  4a;  =  number  of  yards  of  enclosure, 
And  x'"=  number  of  yards  of  pavement, 
Hence4.TX5=  20.r=  price  of  enclosing  it. 
And       x^X  2— 2j2=  the  price  of  the  paving. 
Therefore  2x^ — 20ar,  by  the  question, 

Or  2a;=20,  and  x—  10,  the  length  of  the  side  required. 

6.  Out  of  a  cask  of  wine,  which  had  leaked  away  & 
third  part,  2  i  gallons  were  afterwards  drawn,  and  the  cask 
being  then  guaged,  appeared  to  be  half  full ;  how  much 
« it  hold? 

Let  x=  the  number  of  gallons  the  case  is  supposed  to 
have  held. 

Then  it  would  have  leaked  away  -x  gallons. 

3 


jg2  SIMPLE  EQUATIONS. 

Whence  there  had  been  taken  out  of  it,  altogether, 

21  +  -X  gallons, 
3 

And  therefore  2l4-gX=-a-  by  the  question, 

That  is  63+a=-x,  or  l26  +  2x=3ar, 

Consequently  3a;-2x-=126,  or  3c=126,  the  number  ot 

gallons  required.  .  •  .    r  i 

7.  What  fraction  is  that,  to  the  numerator  ot  which  it  1 

be  added  its  value  will  be  -,  but  if   1  be  added  to  the   de 

o 

nominator,  its  value  will  be  -. 

Let  the  fraction  required  be  represented  -, 

a-4-1      1         .    X        1    .      , 

Then =5.  and  -— -  --,  by  the  question. 

y        3         y-{-l     ^ 

y-\-l 
Hence  3x-\'3—yy  and  4x—y-\- 1,  or  2=—-, 

Therefore  3  ^'^-iL^^  +3=y,  or  3y-^S-\-\2=4y. 

That  IS  y  =  l5,  and  x=~—=  — - — =-r=4, 

4  4  4 

4 
Whence  the  fraction  that  was  to  be  found  is  — ^. 

lo 

8.  A  market  woman  brought  in  a  certain  number  of  eggs- 

at  2  a  penny,  and  as  many  others  at  3  a  penny,  and  having. 

sold  them  out  again,  altogether,  at  the   rate  of  5  for  2d., 

found  she  had  lost  4c/.  ;  how  many  eggs  had  she  ? 

Let  a:  =  the  number  of  eggs  of  each  sort. 

Then  will -a;  =  the  price  of  the  first  sort, 

And  -a:=  the  price  of  the  second  sort, 

4x 
But  5  :  2  : :  2x  (the  whole  number  of  eggs) :  — . 


fcJIMPLE  EQUATIONS.  123 

Whence  — =  the  price  of  both  sorts,  when  mixed  toge- 

:her  at  the  rate  of  5  for  id. 

And  consequently  ^^-r^r — T"""*?  "Y  the  question, 
Z        S         o 

That  is   15f+10x— 'Mx  =  liJO,  or  a:=l2u,  the   number 

of  eggs  of  each  sort,  as  required. 

9.  If  A  can  perform  a  piece  of  work  in  10  days,  and  e 
in  13 ;  in  what  time  will  they  finish  it,  if  they  are  both  set 
about  it  together  ? 

Let  the  time  sought  be  denoted  by  x, 

cc 
Then  —  =  the  part  done  by  a  m  one  day, 

X 

And  :,-^=  the  part  done  by  b  in  one  day. 
lo 

X  X 

Consequently  ^+-=1  (the  whole  work.) 
That  is  13t-{-10x=130.  or  2:^x  =1^50. 
Whence  x=—-=o~  days,  the  time  required. 

10.  If  one  agent  a,  alone,  can  produce  an  effect  c,  in  the 
time  a,  and  another  a^ent  b,  alone  in  the  time  6  ;  in  what 
time  will  both  of  them  together  produce  the  same  effect? 

Let  the  time  sought  be  denoted  by  x. 

Then  a  :  c  :  :  x  :  — =  pari  of  the  effect  produced  by  a. 

€X 

And  b  :  e  :  :  X  :  —=  part  of  the  effect  produced  by  b. 

€X         6X 

Hence j---=e,  (the  whole  effect)  by  the  question, 

d        o 

X       X 

Or  — {-r-=  1  by  dividing  each  side  by  e. 

CLX 

Therefore  x-f  T-=flj  or  bx-^-ax—ab, 

0 

Consequently  3:=— -pi  =  time  required. 
1 .  How  much  rye  at  45.  6d.  a  bushel,  must  be  mixed 


124  SIMPLE  EQUATIONS. 

with  50  bushels  of  wheat,  at  6s.  a  bushel,  so  that  the  uua. 

ture  may  be  worth  55.  a  bushel  ? 

Let  a  =  the  number  of  bushels  required, 
Then  9.r  is  the  price  of  the  rje  in  sixpences, 
And  600  the  price  of  the  wheat  in  ditto, 
Also  (504-j)  X  10  the  price  of  the  wheat  in  ditto, 
Whence  9j-l-600=:500-f- lOr,  by  the  question, 
Or,  by  transposition,  10x~9x=^600-  oOO. 
Consequently  x=lOO  the  number  of  bushels  required: 

12.  A  labourer  engaged  to  o^rve  for  40  days,  on  con- 
dition that  for  every  day  he  worked  he  should  receive 
20d.,  but  for  every  day  lie  was  abst  nt  he  should  forfeit 
S(/.  :  now  at  the  end  of  the  time,  he  had  to  receive  l^ 
1  ]s.  Sd. ;  how  many  days  did  he  work  and  how  many  was 
he  idle  ? 

Let  the  number  of  days  that  he  worked  be  denoted  by 

Then  will  40 — a;  be  the  number  of  days  he  was  idle. 
Also  20-  the  sum  earned,  and  (  lO—  -    X8. 

Or  ^20—8     the  sum  forfeited, 
Whence  20x— (320 -8r)=^380r/,    (=:1/.  lis.  8^.),  b> 
?he  question, 

That  is    Ox— 320+8  i  =  380, 
Or      28.^880H-  .20-^  700, 

Consequently  a-  =       -=   26,   the   number  of  days   he 

worked,  and  40 — .r— 40 — 25=  15,  the  number  of  days  he 
T\'as  idle. 

Q,UF  .STIONS  FOR   PRACTICE. 

1.  It  is  required  '.'•  divide  a  line,  of  "15  inches  in  length, 
into  two  such  parts,  that  one  maybe  thiee  fourths  of  th< 
other.  Ans.  8^  and  '  -f. 

2.  My  purse  and  money  together  are  worth  20s.,  and 
the  money  is  worth  7  times  as  much  a;?  the  purse,  how 
much  is  there  in  it  ?  •  Ans.  175, 6rf. 

3.  A  shepherd  being  asked  how  many  sheep  he  had-  ip. 


SIMPLE  EQUATIONS.  125 

his  flock,  said,  if  I  had  as  many  more,  half  as  many  more, 
and  7  sheep  and  a  hah",  1  should  have  just  600  ;  how  ma- 
ny had  he  ?  Ans.  197. 

4.  A  post  is  one  fourth  of  its  length  in  the  mud,  one 
third  in  the  water,  and  10  feet  above  the  water,  what  is 
its  whole  length  ?  Ans.  2i  teet. 

5.  After  paying  away  -  of  my  money,  and  then  -  of  the 

4  5 

remainder,  I  had  72  guineas  left  ;  what  had  I  at  first  ? 

Ans.  120  guineas. 

6.  It  is  required  to  divide  3«>oZ.  between  a,  b,  and  c, 
so  that  A  may  have  twice  as  much  as  b,  and  c  as  much  as 
A  and  B  together.  An^^.  a  lOOi.  b  60/.  c  150/. 

7.  A  person,  at  the  time  he  ^vas  married,  was  3  times 
as  old  as  his  wife  :  but  after  they  had  lived  together  15 
years,  he  was  only  twice  as  old  ;  what  v/ere  their  ages  on 
the  wedding  day  ? 

Ans.  Bride's  age  15,  bridegroom,  45. 

8.  What  number  is  that  fronn  which,  if  5  be  subtracted, 
wo  thirds  of  the  remainder  will  be  40  ?  Ans.  65. 

9.  At  a  certain  election,.  121^8  p^irsons  voted,  and  the 
successful  candidate  had  a  majority  of  j20  ;  how  many 
voted  for  each  ? 

Ans.  708  for  one,  and  588  for  the  other. 

10.  A's  age  is  double  of  b's,  and  b's  is  triple  of  c's, 
and  the  sum  of  all  their  ages  is  14 )  ;  what  is  the  age  of 
each  ?  Ans.  a's  84,  b's  42,  and  c's  14. 

11.  Two  persons,  a  and  b,  lay  our  equal  sums  of  mo- 
ney in  trade  ;  a  gains  l2o7.  and  b  loses  87/.,  and  a's  money 
is  now  double  of  b's  ;  what  did  each  lay  out? 

Ans.  300/. 

12.  A  person  bought  a  chaise,  horse  and  harness,  for 
60/. ;  the  horse  came  to  twice  the  p»ice  of  the  harness, 
and  the  chaise  to  twice  the  price  of  the  horse  and  har- 
ness ;  what  did  he  give  for  each  ? 

Ans.  13/.  Qs.  8rl.  for  the  horse,  61.  ISs.  4d. 
for  the  harness,  and  40/.  for  the  chaise. 
33-  A  person  was  desirous  of  giving  Sd  apiece  to  some 
i^2 


126  yiMPLE  EQUATIONS. 

beggars,  but  found  he  had  not  money  enough  in  his  pocke- 
hy^Sd.,  he  therefore  gave  them  each  2d.,  and  had  then  3t/ 
emaining  ;  required  the  number  of  beggars  1 

Ans.  11, 

14.  A  servant  agreed  to  Hve  with  his  master  for  8/.  a 
/ear,  and  a  Hvery,  bin  was  turned  away  at  the  end  oi 
seven  months,  and  received  only  2/.  13s.  4d.  and  his  hvery  ; 
ivhat  was  its  value  ; 

Ans.  4/.  16i. 

15.  A  person  left  560/.  between  bis  son  and  daughter, 
n  such    a   manner,    ihat  for   every   half    crown   the  son 

ijhonld  have,  the   daughter  was  to  have  a  shilling  ;  what 
were  their  respective  shares  t 

Ans.  Son  400Z.,  daughter  160/. 

16.  There  is  a  certain  number,  consisting  of  two  placet 
of  figures,  which  is  equal  to  four  times  the  sum  of  its  di 
gits  ;  and  if  18  be  added  to  it  the  digits  will  be  inverted  : 
what  is  the  number  1  Ans.  24. 

17.  Two  persons,  a  and  b,  have  both  the  same  income  : 
A  saves  a  fifth  of  his  yearly,  but  b,  by  spending  50/.  per 
annum  more  than  a,  at  the  end  of  four  years,  finds  himseli 
it  00/.  in  debt :  what  was  their  income  1 

Ans.  125/. 

18.  When  a  company  at  a  tavern  came  to  pay  theii 
reckoning,  they  found,  that  if  there  had  been  there  person.- 
more,  they  would  have  had  a  shilling  a  piece  less  to  pay. 
and  if  there  had  been  two  less,  they  would  b.ave  had  a  shil- 
(mg  a  piece  n;ore  to  pay  ;  required  the  number  of  persons, 
and  the  quota  of  each? 

Ans.  12  persons,  quota  of  each  5s. 

19.  A  person  at  a  tavern  borrowed  as  much  money  as^^ 
liG  had  about  him,  and  out  of  the  whole  spent  Is.  ;  he 
then  went  to  a  second  tavern,  where  he  also  borrowed  as 
much  as  he  had  now  about  ifim,  and  out  of  the  whole 
spent  Is.;  and  going  on,  in  this  manner,  to  a  third  and 
tourth  tavern,  he  found,  after  spending  his  shilling  at  the 
Matter,  that  he  had  nothing  left  ;  how  miich  money  had  he 
'^t  first?  Ans.  U}d. 

vO.  It  is  required  to  divide  the  number  75  into  two  sucl 


SIMPLE  EQUATIONS,  I2T 

pSLXis,  that  three   times  the  greater  shall  exceed  seven 
times  the  less  by  15.  Ans.  64  and  21 . 

21.  In  a  mixture  of  British  spirits  and  water,  ^  of  the 
whole  plus  25  gallons  was  spirits,  and  ^  part  minus  5  gal 
Ions  was  water  ;  how  many  gallons  were  there  of  each  1 

Ans.  85  of  wine,  and  35  of  water, 

22.  A  bill  of  120/.  was  paid  in  guineas  and  moidores. 
and  the  number  of  pieces  of  both  sorts  that  were  used 
were  just  100  ;  how  many  were  there  of  each,  reckoning 
the  guinea  at  21.'<.,  and  the  moidore  at  27s.  1  Ans.  50. 

23.  Two  travellers  set  ont  at  the  same  time  from  Lon- 
don and  York,  whose  distance  is  197  miles  :  one  of  them 
goes  14  miles  a  day,  and  the  other  16  ;  in  what  time  will 
they  meet  1  Ans.  6  days  13|  hours. 

24.  There  is  a  fish  whose  tail  weighs  9lb.,  his  head 
iveighs  as  much  as  his  tail  and  half  his  body,  and  his  body 
weighs  as  much  as  liis  head  and  his  tail  :  what  is  the  whole 
weight  of  the  (ish.  Ans.  72/6. 

25.  It  is  required  to  divide  the  number  10  into  three 
such  parts,  that  if  the  first  be  multiplied  by  2,  the  se- 
cond by  3,  and  the  third  by  4,  the  three  products  shall  be 
all  equal.  Ans.  4,-83,  3yV»  ^t\' 

2G.  It  is  required  to  divide  the  number  36  into  three 
such  parts,  that  1  the  first,  i  of  the  second,  and  |  of  the. 
third,  shall  be  all  equal  to  each  other 

Ans.  The  parts  are  8,  12,  and  16. 

27.  A  person  has  two  horses,  and  a  saddle,  which  ot 
itself  is  worth  50/.  ;  now,  if  the  saddle  be  put  on  the 
back  of  the  first  horse,  it  will  mak  e  his  value  double  that 
of  the  second,  and  if  it  be  put  on  the  back  of  the  second^ 
it  will  make  his  value  trii)le  that  of  the  first  ;  what  is  the 
value  of  each  horse  1        Ans.  One  80/.  and  the  other  40/.. 

28.  If  A  gives  B  5s.  of  his  money,  b  will  have  twice  as 
much  as  the  other  has  left ;  and  if  b  gives  a  5s.  of  his  mo- 
ney, A  will  have  ihrcc  times  as  much  as  the  other  has  left : 
iiow  much  had  each  ?  Ans.  a  13s.  and  b  1 1*. 

29.  What  two  numbers  are  those  whose  difference, 
sum  and  product,  are  to  each  other  as  the  numbers  2,  3, 
nnd  5  respectively  1  Ans.  10  and  2, 


:28  QUADRATIC  EQUATIONS. 

30.  A  person  in  play  lost  a  fourth  of  his  money,  and 
then  won  back  3^.,  :ifier  which  he  Njst  a  third  of  what  he 
now  had,  and  thnti  won  back  2s.  ;  lastly,  he  lost  a  seventh 
of  what  he  then  had,  and  after  this  found  he  had  but  I  is. 
remaining  :   what  had  he  at  Hr<t  ?  Ans.    Os. 

3i.   A  hare  is  50  !«aps  before  a  greyhound,  and  takes 

4  leaps  to  the  ^rreyhoutid's  ^,  but  2  of  the  greyhound's 
leaps  are  as  much  as  3  of  the  hare's;  how  many  l^aps 
must  the  greyiiound  t<ike  to  catch  the  hare  ?  Ans.  300. 

^^2.  It  is  required  to  divide  the  number  90  into  tour 
such  parts,  that  if  the  first  part  be  mcreased  by  2,  the  se- 
cond diminished  by  2,  the  third  multiplied  by  2,  and  the 
fourth  divided  by  ^,  the  sum,  difference,  product,  and 
•{uotient.  shall  be  all  equal  ? 

Ans.  The  parts  are  18,  22,  10  and  40. 

33.  There  are  three  numbers  whose  diflerences  are 
equal,  (that  is,  the  second  exceeds  the  first  as  much  as  the 
third  exceeds  the  second),  and  the  first  is  to  the  third  as  5 
to  7  ;  also  the  sum  of  the  three  numbers  is  324,  what  arc 
those  numbers  ?  Ans.  ^.O,  1  08,  and  126. 

34.  A  man  and  his  wife  usually  drank  out  a  cask  of  beer 
in  12  days,  but  when  the  man  was  from  home  it  lasted  the 
woman  30  days  :  how  many  days  would  the  man  alone  be 
in  drinking  it?  Ans.  20  days. 

35.  A  general  ranging  his  army  in  the  form  of  a  solid 
square,  finds  he  has  -.^84  men  to  spare,  but  increasing  the 
side  by  one  man,  he  wants  25  to  fill  up  the  square  ;  how 
many  soldiers  had  he?  Ans.  i'4000. 

:i6.  If  A  and  b  together  can  perform  a  piece  of  work  in 

5  days,  a  and  c  together  in  9  days,  and  b  and  c  in  10  days, 
iiow  many  days  will  it  take  each  person  to  perform  the 
same  work  alone. 

Ans.  A  14fi  days,  e  17|f,  andc  23/-i. 

QUADRATIC  EQUATIONS. 

A  Quadratic  EauATioN,  as  before  observed,  is  that  in 
v/hich  the  unknown  quantity  is  of  two  dimensions,  or  which 
rises  to  the  second  power,  and  is  generally  divided  ixiU 
simple  or  pure,  and  compound  or  adfected^ 


\ry 


QUADRATIC  EQUATIONS.  129 

A  simple  or  pure  quadratic  equation,  is  that  which  con^ 
lains  only  the  square,  or  second  power,   of  the  unknown 

quantity,  as 

aa;^--=6*  or  a-^-|--  j  where  T  =  y/-. 
a  CL 

A  compound  or  adfected  quadratic  equation,  is  that 
which  contains  both  the  first  and  second  power  of  the  un- 
known quantity,  as 

2    1/.  "I    *  ^ 

a       a 
In  which  case  it  is  to  be  observed,  that  every  equatioii 
of  tliis  kind,   having  any  real  positive  root,   will  fall  under 
one  or  other  of  the  three  following  forms  : 

'^  x^-\-ax=^b  .  .  .  where  J = — -±:-)/\—--{-b). 


*  It  may  be  obsprved,  with  respect  to  these  forms,  that 

In  the  case  x'i  ^ax—brr^O,  where  i  =: — iix  T-v^(ia3  4.6),  or— ia- • 
^(J-a2-^6),  ihe  first  value  of  ic  must  oe  posiiive,  because  v  (ia2 -f.  6)  is 
greater  than  y/  \a2,  or  its  equal  ^a,  <.iid  us  second  value  will  evidently  be 
negative,   because  each  of  the  terms  o;  w  mK  it  i?  composed  is  negative. 

2.  In  the  case  ,r2 — ax — b  z=>o,  where  x  =s^a^  y/(^as  ^b)ar  ^a—-^ 
(^a2  -f-fc),  the  first  vahie  of  r,  is  maitil«>*ll\  positive,  bemg  the  sum  of  two 
positive  terms:  and  ihe  -econd  value  vv>.:  be  nogalivc,  because  'y/(ya2  -J-6) 
is  greater  than  -y/  \a2    or  its  equal  ^  a. 

3  Inthecasea;2  —  ic-f-6=30,  \vi;ere  a  -ja-f-vfdaS — 6),  or  ^a — y'" 
(ia2 — 6),  both  the  Viiues  of  a;  will  be  positive,  wlien  'ia2  is  greater  than  b  ; 
for  its  first  value  is  'hen  evidently  positive,  beiiii^  composed  of  two  positive 
terms;  and  its  se;ond  value  will  also  be  positive;  berause\/(^a2— 6)  is 
less  than  \/ ^aS ,  or  its  equal  ^a. 

But  if  ja2 ,  in  this  case  be  less  than  b,  the  solution  of  the  proposed  equa- 
tion IS  impossible;  bt:cau*e  the  quantity  4-02 — b,  under  the  radical,  is  then 
negative;  and  consequently  V  ( j«2— 6)  wil!  be  imaginary,  or  of  no  assign- 
able value. 

4.  It  may  be  also  further  observed,  that  there  is  a  fourth  case  of  the  form 
x2  ^aX'^bs=o,  where  a;s= — ^aj^^/ijas — 6),  or  x—'^a—^  (^a2 — 6), 
the  two  values  of  >  will  be  both^negativ^,  or  both  imaginary,  according  as 
;^a5  is  greater  or  less  than  6 ;  the  imaginary  roots,  when  they  occur,  being 
liere  of  the  forms — (a'  -|-c'^  — 1)  and — («' — c.'^ — 1). 

From  which  it  follows,  that  if  all  the  'erms  of  a  quadratic  equation,  when 
brought  to  the  left  hand  side,  be  positive,  its  two  roots  will  be  both  negative, 
or  both  imaginary  and  conversely,  if  each  of  the  roots  be  negative  or  eaci, 
imaginary,  the  signs. of  all  the  term?  will  be  positive. 

So  that  of  all  quadratic  equations,  which  can  have  any  r  'al  positive  root, 
ihat  of  the  third  form,  x2 — a^-j-  b  =0,  is  the  only  one,  where  the  solutiot. 
for  certain  nuntieral  values  of  a  and  b,  will  beccme  Impossible 


130  QUADRATIC  EQUATIONS. 

2.  X — tLx^h   .   .   .   where  x=-h-±.x/\—'   hbj . 

3.  a;2—ai-=— 6  .   .   where  x  =  -^^:^  y(°^- b). 

Or,   if  the  sec  >«'!  and   la-?t  terms  he  taken  either  posi 

lively  or  negatively,  as  they  may  happen  to  be,  the  genera! 

equation 

^      h  c 

ax^±.hx—  He,  or  x^^:  -r—  t- 

'i  a 

which  comprehends  all  the  threM  cases  above  mentioned, 

rnay  be  resolved  by  m'^ans  of  the  following  rule  : 

RULE. 

Transpose  all  the  terms  that  involve  the  unknown  quan- 
tity to  one  side  of  th.^  f^qmtiori,  and  the  known  terms  to 
the  other;  observinjj  «o  arran^ze  them  so  that  the  term 
which  contains  the  sqa-ire  )«"  tue  .mknown  quantity  may 
be  positive,  and  stand  hrst  in  the  equa  ion. 

Then,  if  this  square  has  any  r ^efficient  prefixed  to  it, 
let  all  the  rest  of  'he  terms  be  divided  by  it,  aiui  the  equa- 
tion will  be  broutjht  t  >  one  of  tne  three  forma  above- 
mentioned. 

In  which  case,  the  value  of  the  unknown  quantity  a:  is 
always  equal  to  half  the  coeffii-icnt,  or  mullipher  of  x,  in 
the  second  term  of  the  equation,  taken  with  a  contrary 
sign,  together  with  +  the  -^qiare  root  of  the  square  of  this 
number  and  the  known  quantity  that  forms  the  absolute  or 
third  terra  of  the  equation.* 


*  This  rule,  which  is  more  commodious  in  its  practical  application  than 
■ijal  usually  given,  is  founded  upon  the  same  principle;  being  derived  from 
the  well  known  p'operty,  that  in  any  quadraiic 

a2   -rax  =  x  ^'  '^ ''^^  "^quar*-  of  half  the  coeflBcient  a 
of  the  second  term  of  the  equation  be  ndded  to  each  of  its  sides,  so  as  to  ren- 
der it  of  the  form  , 

Iiat  side  which  contains  the  unkccwj  quantity  will  then  be  a  complete  square  . 
ind,  consequently,  by  extraciins;  the  '•ootof  each  side,  we  shall  have 


QUADRATIC  EQUATIONS.  13  J 

Note,  All  equation  J,  which  have  the  index  of  the  un- 
known quantity,  in  one  vti  their  terms,  just  double  that  ot 
the  other,  are  resolved  Uk«  quadratics,  by  first  finding  the 
value  of  the  square  root  of  the  first  term,  according  to  the 
method  used  in  the  above  rule,  and  then  taking  such  a  root, 
or  power  of  the  result,  as  is  denoted  by  the  reduced  index 
of  the  unknown  quantity. 

Thus,  if  there  be  taken  any  general  equation  of  this  kind, 
as, 

"a--"-  -f  aa-"*=6, 
we  phall  have,  by  takins;  the  square  root  of  x-*",  and  observ- 
ing the  latter  part  of  the  luie, 

And  if  the  equation,  \vhich  is  to  be  resolved,  be  of  the 
following  form, 

m 

X"*  —fix^=bf 
ve  shall  necessarily  have,  according  to  the  same  principle. 

.»-"=|±v/(^+*),  and  x=  I l+^{^+b)  j- 

EXA5IFLES. 

1.  Given  a:^-|-*4a;=140,  to  find  the  value  of  a;. 
Here  x^-\-'\x=l'iO^  by  the  question. 
Whence  a;=— 2  + v  (4+140),  by  the  rule, 

Or,  which  is  the  same  thing,  x=  -^±:y^l44. 


which  is  the  same  as  the  rule,  taking  a  and  6  in  -f  or  —  as  they  may  haj 
pen  to  be. 

It  may  here  also  be  observed,  that  the  ambiguous  sign  ^,  which  denotes 
both  -j-  and — ,  is  prefixed  to  the  radical  part  of  the  value  of  x  in  every  ex 
pression  of  ihis  kind,  because  the  square  root  of  any  positive  quantity,  as  a2 
is  either  -*-  a  or  — a  ;  for  (-f-a)  X(  r  a),  or  (—a)  x  ( — «)  are  each  =  -J" 
•2  ■  but  the  square  ror  t  of  a  negative  quantify,  as  — au ,  is  imaginary,  or  un- 
assignable, there  being  no  quantity,  either  positive  or  negative,  that  when 
multiplied  by  itself,  wil!  give  a  negative  product 

To  this  we  may  also  turther  add,  that  from  the  constant  occurrence  of  the 
double  sign  before  the  radical  part  of  the  above  expression,  it  necessarily  fo!- 
lows,  that  every  quadratic  equation  must  have  two  roots  ;  which  are  either 
both  real,  or  both  imaginary,  according  to  the  nature  of  the  rjuestion. 


i32  QUADRATIC  EQUATIONS. 

Wherefore  a;=  —  24-l-J  =  10,  or  —2-  12=—  14. 
Where  one  of  the  values  of  x  is  positive  and   the  other 
negative. 

2.  Given  af  ~  l2a:-}-30=3,  to  find  the  vahie  of  .r. 
Here  x^ -]2x='i -3v  ~  ~  27,  by  transposition. 
Whence  x=6  iV("-^'>  -27),  by  the  rule, 

Or,  which  is  the  same  thing  x=b^y'9, 
Therefore  x= 6 -{-  3  — -  9,  or  —  6 — 6  =3. 
Where  it  appears  that  x  has  two  positjvp  values. 

3.  Given  2a; '+8a,— 20=^70,  to  hricl  ihi  value  of  .r. 
Here  2a;^4-Ha?=  ■  0-t-20=b(),  by  tiansposition. 
And  x^-i-4x~^5y  by  cividing  by  2, 

Whence  a;=-2-   V    » 4-^5),  by  the  rule, 
Or,  which  is  the  sarne  thmg,  .t— — :^+ -x/l9. 
Therefore  a;=  — .-[-7=5,  or  -  -.2—.     — 9. 
Where  one  of  the  values  of  x  is  positive  and  the  oth^» 
negative. 

4.  Given  3x'— 3a;+6=  H,  to  find  the  value  of  .r. 

Here  Saf—Sx^b^-^e^—^  by  transposition. 

2 

And  x^  -  x=  -  -  by  dividing  by  3, 

Whence  '^=2— '^(i^g)'  ^y^^^rule. 
Or,  by  subtracting  -  from  .,  «=-+y/_- 

-ru       c  *  .  1      2  III 

1  nereiore  x= — — =-,  or  = =-, 

2^6     3'  2     6     3' 

In  which  case  x  has  two  positive  values. 

5.  Given -a?2—-  a;+20±=42f  to  find  the  value  of  a. 

Here  g  «'  -  3  x=4^— 20^=22}  by  transposition, 

2  1 

And  .r^—- a;  =441,  by  dividing   by-,  or   multiplier 

Whence  we  have  x=^±^/(g+^^),  by  the  rtile. 


QUADRATIC  EQUATIONS.  ISS 

t)r,  by  adding  -  and  44|  together,  a;=^±  ^J  — -, 

Therefore  a:=i-f-6|=7,  or  =rl-6|=-6i, 

Where  one  value  of  x  is  positive  and  the  other  ne 
gative. 

6,  Given  ax^-\rhx—Cj  to  find  the  value  of  t« 

h       c 
Here  x^-i — x=-  by  dividing  each  side  by  a. 

Whence,  by  the  rule.  x=_^±v(— +^). 

Or  multiplying  c  and  a  by  4a,  a;=  —  ^±^ — T's" 

Therefore  x=— ^±i-v/(62-f  4ac). 

7.  Given  ax^'~bx-{-c=d,  to  find  the  value  of  a% 
Here  ax^'~bx=^d  -c,  by  transposition, 

And  x^ — x= by  dividing  by  a. 

Whence  ^=?r-±v/(  — - — I — o^  by  the  rule, 

'    Or,  multg  d—cSaa  by  4a,  x=—+^y/{4a(d^c)-\'b\ 

3.  Given  x^-{-ax^=hi  to  find  the  value  of  a:. 
Here  a:'*+ 0x^=6,  by  the  question, 

Orx^=-^±^(^+6)=-|±iv'(a^+46),     by     the 

rule, 

Whence  a:=  +  y (  — ^^:-v^ (46  H-a^))  by  extraction  o* 

roots. 

9.  Given  -x^ — x^=  — 5jr,  to  find  the  value  of  s" 

Here  -x^~  -x^=  -  — ,  by  the  question, 

And  ar6-.-a;3=:^       by  multiplying  by  2, 
.V  lb 

N 


34  QUADRATIC  EQUATIONS, 

Whence  a:'^=^-±v'(-^— ^),=-  by  the  rule, 

1         2     1 

And  consequently  a=y-=y-=-V'2. 

10.  Given  223  4-3x3=2,  to  find  the  value  of.r. 

2  1. 

Here  2x3  4-3x3  =2,  by  the  question, 

2       3    i 

And  a;^+-x^  =  l,  by  dividing  by  2,    . 

til 

Whence  .==-|±^(^+l)=-5±54  or -2 
Therefore  as=(l)»=i  or  (_2)'=-8. 

tit  o 

11.  Given  x^— 12x3 4-44a;2-48x=  9009(a),  to  find  the 
value  of  X. 

This  equation  raav  be  expressed  as  follows, 

*(x^-Dx)2+8(x=— 6x-)=a, 

*  The  biquadratic  equation 

a:^— 122;'4-44i-— 43xc=  a 
can  be  easily  exhibited  under  the  form 

(x^— 6.T)^-f  8(x='— 6i)  =rf 
hy  the  following  method. 

o;^— 12x3  4-44x2  _48r(a:2— Sx- 


21^— 6i)— 12a  3  4-442*--48x 
-12^3-1-36^2 


-6xj  Sr*— 48x(R 

8a;2— 48r 


Ck)nsequently,  \x^ —^xf -^  %[x^ —^x-)^  x"^ —\i%^  ■\.^^x^ —^%x  z=  a  : 
for  since  in  extractinsf  the  square  root  of  at;}'  quantity,  the  J.quare  ot  the  root 
'hus  found  plus  the  remainder  is  always  equal  lo  tiie  proposed  quantity. 

In  a  similar  manner,  the  biquadratic  equation  a^  -LSar"  -|-5a^x^4-4fi" 
f  =  d,  tnay  be  exhibited  under  the  form 

vybich  can  be  resolved  by  the  rule,  page  130,  for  resolving  quadratic  equa- 
tions. 

Hence  it  follow?,  that  if  Ihe  remainder,  after  having  found  the  nrst  two 
'erms  cf  (lie  square  root,  according  to  the  rule  ^ij^je  49,  can  be  resflvedinto 


QITADKATIC  EQUATIONS.  156 

Whence  x^-'Qx=^  -.4±y^(l64-a),  by  the  common  rule, 
And,  by  a  second  operation,  x'=3^r-v/(9— 4±<v/(164-a)) 

Therefore,  by  restoring  the  value  of  a,  we  have 
a;=3±^(5±^9025) 

Or,  by  extraction  of  roots,  a:=^l3,  the  Ans. 

EXAMPLES  FOR  PRACTICE.* 

1.  Given  x^—  8a; -f  10=19,  to  find  the  value  of  x. 

Ans.  a=9, 

2.  Given  a:^— or— 40=170,  to  find  the  value  of  x. 

Ans.  a;=l5 

3.  Given  Sx'''\-2x  «  9=76,  to  find  the  value  of  a-. 

Ans.  a; =5-. 

4.  Given  ^a;^— ^a^+Tf =8,  to  find  the  value  of  x. 

Ans.  z=-\\. 

5.  Given  ^x  — -^x=22^,  to  find  the  value  of  x, 

lit  o 

Ans.  x=49. 

6.  t  Given  x-f-\/(5x-{-l0)=8,  to  find  the  value  of  x. 

Ans.  x=3. 

two  such  factors,  so  that  the  factor  containing  the  unknown  quantity,  shall 
be  equal  to  the  terms  of  the  root  thus  found  ;  the  proposed  biquadratic  may 
be  always  reduced  to  a  quadratic  form,  as  above.  See  Ryan's  Algebra^pag* 
396.  E^. 

*  The  unknown  quantity  in  each  of  the  following  examples,  as  well  as  in 
those  given  above,  has  always  two  values,  as  appears  from  the  common  rule; 
but  the  negative  and  imaginary  loots  being,  in  general,  but  seldom  used  ia 
practical  questions  of  this  kind,  are  here  suppressed. 

+  In  some  quadratic  equations  involving  radical  quantities  of  the  form  V' 
(aa;-f-6),  both  the  values  of  x  found  by  the  ordinary  process,  will  notanswet 
the  proposed  equation,  except  we  take  the  radical  quantity  with  the  double 
sign  -^ .  In  resolving  the  above  exantple,  two  values  of  x,  that  is,  18  and  3, 
are  found  ;  but  it  appears,  that  18  does  not  answer  the  condition  of  the  equa- 
tion except  we  lake  the  radical  quantity  ^yiSx  -\-  10)  with  the  sign  — . 

Now,  since  these  two  values  of  x  are  formed  from  the  resolution  of  the 
equation  x2  — 21  t= — 54 ;  it  necessarily  follows  that  each  of  them,  when  sub* 
stituted  for  x,  must  satisfy  that  equation  ;  which  may  be  verified  thus ;  in  the 
first  place,  by  substituting  1 8  for  x,  in  the  equation  x2  — 21  x  =—54,  we  have 
(18)2— li  X  18  =  — 54,  or  324— 378:=— 54,  that  is,— 64^— 54,  or  by 
transposition  0=0. 

Again, substituting 3  for  x,  we  have  (3)2—21  X  3  =—54, or 9 — 63=— 54; 
.  .51— 14-=0,  orOi?=0. 

And  as  the  equation  x2 — 21x=: — 54.  may  be  deducted  from  the  equation 
4- \/(5a;-|-  10)  =  8— x,or  —  v/(5x+10)  s=  8— or ;  it  is  evident  that  the  radU 
cal  qiiantitj  v^(5x-f-10)  must  be  taken  with  the  doubl.esigQ  hk,  in  the  pri- 


It6  QUADRATIC  EQUATIONS, 

7.  Given  y/{\0'hx)'-\/{\0+x)=2,  to  find  the  valuT 
of  ar.  Ans.  x=6. 

8.  Given  2z*-ar2-f96=99,  to  find  the  value  of  x. 

Ans.  x=^^6 

9.  Given  a;*+20a;3—  10=59,  to  find  ^e  value  of  a;. 

Ans.  x=\/S. 

10.  Given  3ic^— 2x"+3=l  1,  to  find  the  value  of  x, 

Ans.  a;=y2. 

11.  Given  6\/x — 3*/a?=l^»  to  find  the  value  of  x. 

3 

Ans.  3- or-. 

o  12 

1^.  Given  o^'v/(3-|-2a;2)=--|--a;^  to  find  the  value  of  .r-. 

Ans.  a:-i^( -3-1-3^/2), 

13.  Given  a:v/'(--a?W-^t^  to  find  the  value  of  .r, 

^x       /       ^x 

Ans.  a;=(l4-^v/2)2. 

14.  Given  -v/(l— a;^)=a;2,  to  find  the  value  of  x, 

X 

Ans.  x=Q^5^^-)\ 

15.  Given  «./(-- l)—^(a?=—62^,  to   find   the   value 
of  X,  Ans.  a;=ia4— -v/(86H«')- 

16.  Given  v'(l-fx-a;=)  •<2(  1+0^—072)=^  ^^  ^^^^  ^l^^ 

value  of  x»    "  Ans.  a;— -4--v/41. 

^     o 

milive  equation,  in  order  that  it  would  be  satisfied  by  the  values,  18  and  S, 
of  X,  found  above  ;  that  is,  18  ansvvers  to  the  sign  — ,  and  3  to  the  sign  -f-- 
See  Ryan's  Elementary  Treatise  on  Algebra,  Theoretical  and  PracticaH 
•vhere  thia  subject  is  clearly  illustitited.  Ef'. 


QUADRATIC  EQUATIONS\  m 

17.  Given  ^(x — )+  V  (^  — )  =a;,  to  find  the  valujB 

of  oe.  Ans.  x=^-^^y/5. 

18.  Given  a?^"— 2a;3«-f-a?»=6,  to  find  the  value  of  x. 

Ans.  a;=y(i-f iv^l3). 

19.  Given  a;^— 2a;^-{-a;=a,  to  find  the  value  of  x. 

Ans.x=i=^J?=v(a+l)j. 

When  there  are  more  equations  and  unknown  quantities 
than  one,  a  sinjjle  equation,  involving  only  one  of  the  un- 
known quantities,  may  sometimes  be  obtained,  by  the  rules 
before  laid  down  for  the  solution  of  simple  equations  ;  and, 
in  this  case,  one  of  the  unknown  quantities  being  deter- 
mined, the  others  may  be  found,  by  substituting  its  value 
in  the  remaining  equations. 

EXAMPLES. 

1.  Given  \  ^   '  ^  ~~oc  {   to  find  the  values  of  x  and  v. 
i       xy  =28  ^  ^ 

28 
Here,   from  the  second  equation,   we  have  ?/= — ;   and 

784 
by  substituting  this  in  the  first  x^-\ — ~~  65,    or   a;*— 65a;^ 

c^-784. 
Whence,  by  the   common   rule  before  given,  we  have 

_         ^65  /4225        ^\l 

Or,  by   reducing  the  parts  under  the  last  radical,  and 

/65     33\ 
extracting  the  root  x~±l^/1—±l~-\~1,  or  — 4,  and  con- 

28            28 
.seauenlly,  w= — ,  or =4  or  —  7. 

Or  the  solution,  in  cases  of  this  kind,  may  often  be  more 
readily  obtained,  by  some  of  the  artifices  frequently  made 
lise  of  upon  these  occasions  ;  which  can  only  be  learned 
from  experience:  thus,  taking  as  before,  (1.)  x"4"3/^-=65f 

n2 


.38  QUADRATIC  EQIXATIONS. 

(2.)j:jy=28,  we  shall  have,  as  in  the  former  niathod,  by 
multiplying  by  si,  2xy=56,  and,  by  adding  this  equation  to 
the  first,  and  subtracting  it  from  the  same,  x^-\-':xy-{-y^=: 
121,  and  z^-^Sa;!/ +2/^=9.  Whence  by  extracting  the 
square  roots  of  each  of  these  last  equations,  there  will 
arise,  a;+t/=^il  I,  and  r  -j/=Ht3,  and  consequently  by 
adding  and  subtracting  these  we  shall  have  2x=^\4,  or 
r=7,  or -7,  and  ^=4,  or — 1.  It  will  also  sometimes 
facilitate  the  operation  by  substituting  for  one  of  the  un* 
known  quantities  the  product  of  the  other,  and  a  third  un- 
known  quantity  ;  which  method  may  be  applied  with  advan 
tage,  whenever  the  sum  of  the  dimensions  of  the  unknown 
quantities  is  the  same  in  every  term  of  the  equation. 

2.  Given  }^  X2^-=60  (  *^  find  the  values  of  x  and  j'. 

Here,  agreeably   to  the   above  observation,   let  x=vy, 

then  v"y^-{-vy^=b6f  and  v^^-}-2?/"=60,   whence,   from   thr 

56 
first  of  these  equations,  y^=^-r-, — >   and  from   the    second 
v-f-v 

2/^=     ,     .     Therefore,  by   equating  the  right  hand  mem- 

ber  of  these  two  expressions,  we  shall  have  — -  -—-^, — , 

or  60v^-|-6OT>=56u-f-I12.     And,  by  transposing  56r',  and 

1         3S 

dividing  the  result  by  60,  7/^-1 — -v=— .       Hence    by    the 

common  rule,    for  quadratics,  we  have  tj= — "n"^  v^(  qiT- 

)  28  1,414,^  ,      ,       .      r 

-r7?=^~r7^+57r="--     And,   consequently,   by  tne  lorni- 

o       60  60 

er  part  of  the  process,  y-=-— -  =  — — —  =  1&,  or  y=^ 
v-\-z     1^+2 

4 
(  0       3v/2,  and  x=-yy=-X  3  v' 2=.4v/^.     The    work 

may  also  be  sometimes   shortened,  by  substituting  for  the 
unknown  quantities,  the  smn  and  di0e:ence  of  two  other 


QUADRATIC  EQUATIONS.  139 

c[uantities  ;  which   method  may  be  used,  when  the  un- 
known quantities,  in  each  equation,  are  similarly  involved. 


{x    -hv  =12) 


3.  Given   \  y  "^  x  )to  find  the  values  of  re  and  yi 

Here,  according  to  the  above  observation,  let  there  be 
assumed  x=2r-}-v,  and  y=z—v.  Then,  by  adding  these 
two  equations  together,  we  shall  have  x^y=^z=^\2,  or 
2=6y  also,  since  x—6-\-v,  y=6—v,  and  by  the  first  equa^ 
tion  x^-{-y^=^\3xyj  we  shall  obtain,  by  substitution,  (6-f- 
vy'{-{6  —  vy=l8{Q'{-v)i6 — v),  or,  by  involving  the  two 
parts  of  the  first  member,  and  multiplying  those  of  the 
second,  432-{-3hV=:648— ISr^,  whence,  by  transposition 

54.i;2— o|e  .  and  by  division,  r^— .___— 4  .  qy  v=  ±:2, 

And  therefore,  by  the  first  assumption,  and  the  first 
part  of  the  process,  we  have  x=z-{'V~6±.2=-B,  or  4, 
and  y—z—v—6:ii2=-\,  or  8. 

QUESTIONS  PRODUCING  QUADRATIC 
EQUATIONS. 

The  methods  of  expressing  the  conditions  of  questions 
of  this  kind,  and  the  consequent  reduction  of  them,  till 
they  are  brought  to  a  quadratic  equation,  involving  only 
one  unknown  quantity  and  its  square,  are  the  same  as 
those  already  given  for  simple  equations. 

1 .  To  find  two  numbers  such  that  their  difference  shall 
be  8,  and  their  product  240. 

Let  X  equal  the  least  number. 
Then  will  a-+8=the  greater, 
And  x{x-{-8)=x^+8x='240,  by  the  question, 
Whence  x=-4-f  v^(164-240;  =  -44-v^256  by  the  com> 
men  rule,  before  given. 
Therefore  x=\6 — 4  =  12,  the  less  number, 
and  a;-|-8=  12+8=20,  the  greater. 


;40  QUADRATIC  EQUATIONS. 

2.  It  is  required  to  divide  the  number  60  into  two  sUcl 
parts,  tiiat  their  product  shall  be  864. 

Let  x=  the  greater  part, 
Then  will  60— x=  the  less, 
And  0^(60  — a;)  =  60a:-~x"— 864,  by  the  question, 
Or  by  changing  the  signs  on  both  sides  of  the  equation 

x2-60a:=  — 864, 
Whence   a;=.iO±  y/  (900  -  864)  =30j£v' 36=30+6,  by 

the  rule, 

And  consequently  a;=3o-f6=36,  or  30  —  6=24,  the  two 

parts  sought. 

3.  It  is  required  to  find  two  numbers  such,  that  their 
sum  shall  be  10(a),  and  the  suni  of  their  squares  58(6). 

Let  x=  the  greater  of  the  two  numbers, 

Then  willa  — x=  the  less, 

And  a;-4-(«— ar)-=2x^--2ax=a^=6,  by  the  question, 

Or  2x-  — 2ax=6  —  a^,  by  transposition, 

Andx^— 02;=— -— ,  by  division. 

Whence  x=^±^(^-+—^)=:=-±-^{'ib-o:^) 

by  the  rule, 
:Vnd  if  10  be  put  for  a,  and  58  for  6,  we  shall  have 

a:=— 4-^v/(^  ^^"-  100)=7,  the  greater  number. 

10      I 

And  10  ^  a:=— ---^/(l  16  -  100)=3,  the  less. 

4.  Having  sold  a  piece  of  cloth  for  34/.,  I  gained  as 
much  per  cent,  as  it  cost  me  ;  what  was  the  price  of  th* 
-loth  ^ 

Let  .r=pounds  the  cloth  cost. 

Then  will  24  — a;  =  the  whole  gain. 

But  100  :  a:  : :  rt  :  24 — x,  by  the  question, 

Or  a:-=  100(24— x)  =  2400  -  lOOx, 

Thatis,  x2+100x=2400, 

Whence  a-— 50-1-^/(2500-1-2400)=- 50-r70  ^-  2'- 

by  the  rule. 


QUADRATIC  EQUATIONS.  141 

An9  consequently  20^=price  of  the  cloth. 

6.  A  person  bought  a  number  of  sheep  for  80/,  and  if 

he  had  bought   four  more  for  the  same  money,  he  would 

have  paid  l/.  less  for  each  ;  how  many  did  he  buy? 

Let  X  represent  the  number  of  sheep, 

80 
Then  will  —  be  the  price  of  each, 

X 

SO 
And  — r—=  price  of  each,  if  as+4  coat  80/. 
a;-l-4 

80       80 
But  — =— 7-:  +1»  by  the  question, 
X      a;-f-4 

POt 

Or  80=— r—-  4-x,  by  multiplication. 
a;-t-4 

And  80x+320=80a:-f  ar2+4x,  by  the  same, 
Or,  by  leaving  out  8  Ox  on  each  side,  ;r='-|-4a;=320, 
Whence  a;=-2-f-v'(44-320)=  -2+18,  by  the  rule, 

And  consequently  x=16,  the  number  of  sheep. 

6.  It  is  required  to  find  two  numbers,  such  that  theii 
sura,  product  and  difference  of  their  squares,  shall  be  all 
equal  to  each  other. 

Let  a;=  the  greater  number  and  ^=  the  less. 

Hence  1=— — ^  =.x  — ?/»  or  x=2/+l,  by  '^d  equation. 

And  (y+l)+2/=J/(!/-i-l),  by  1st  equation, 
That  is,  2y+\=f-\ry',  r +  !/—»• 

Whence  2/=^+v/(^+l)=2+^i/5»  by  the  rule, 
Therefore  2/=7-f^v^5  =1.6180.  .  . 

Anda;=^+l=?4-^v'5=2.6180  ... 

Where  .  .  .  denotes  that  the  decmial  does  not  end. 

7.  It  is  required  to  find  four  members  in  arithmetical 
progression,  such  that  the  product  of  the  two  extremes 
^hall  be  45,  and  the  product  of  the  means  77, 


i42  QUADRATIC  EQUATIONS. 

Let  x=-  least  extreme,  and  y=-  common  difference, 
Then  a;,  a:-|-y,  x+2i/,  and  x-h3y,  will  be  the  four  number^^, 

riuestion, 

And  21/2=77-45  =  32,  by  subtraction, 

32 

Or  ^=— =  16  by  division,  and  ^=^16=4, 
ci 

Therefore   a:24-3Ti/=ar2+12a:=45,  by  the   1st  equation.. 

And  consequently  x=  — 64-^/(36-1-46)  =  — 6-j-9=3,   by 

the  rule. 

Whence  the  numbers  are  3,  7,  11,  and  15. 

8.  It  is  required  to  find  three  numbers  in  geometrical 
progression,  such  that  their  sum  shall  be  1 4,  and  the  sum 
of  their  squares  84. 

Let  X,  y,  and  z  be  the  three  numbers. 
Then  xz^^y^,  by  the  nature  of  proportion, 

^"^   )x$i^-f?i=84   \   by  the  question, 
Hence  a: 4-2=  '4 — y,  by  the  second  equation, 
And  a;^+2^x'-{-^2=l96  — 2by4-2/^,     ^V    squaring     both 

sides, 
Or  x-+224.2>/2=19f]  — 28j/-f-2/2  by    putting   2y^   for  its 
equal  2a:^, 
That  is  x-'-f-^^-f  22_  ,  9g  _  ,9>y  by  subtraction, 
Or  i96— 28t/=i84  by  equality, 

.r  »96     84      ^    ,  .  .  J  J.  .  . 

nence  y— — ^     — =4,  by  transposition  and  division. 

1  (\ 

Again  x2'=^2_.|g^  q^.  x-=— ,  by  the  1st  equation, 

I  fi 
And  x-\-y-\'Z^ h4-f-2'  =  14,  by  the  2d  equation, 

Or  16-{-42'4-2'2=:14r,  or^2  -  10^=~16, 
Whence  ^=5jh  ^^(25- I6)  =  5  +  3—-8,  or  2  by  the-rule* 
Therefore  the  three  numbers  are,  2,  4,  and  8. 

9.  It  is  required  to  find  two  numbers,  such  that  theii 
sum  shall  be  13(a),  and  the  sum  of  their  fourth  power? 
4721  (6). 


QUADRATIC  EQUATIONS.  14S 

Let  a=  the  difference  of  the  two  numbers  sought, 

Then  will  qO+^x,  or        '  =  the  greater  number^ 

.     ,  I        1  a — X       ,     . 

And  ^■-  ^a-,  or  — q— =  the  less, 

But  ^- — -| — =6,  by  the  question, 

Or  {a+xY-hia-xY—lGh,  by  multiplication, 
Or  2a^-\-l2a''x^+2x'=l6b,  by  mvolution  and  addition. 
And  aj^+eaV^Si-a",  by  transposition  and  division. 

Whence  x^=^  -  3»"4-x/(9Q^-f  b6  -ft^j=-3a^= 
^8(a^+fe),  by  the  rule, 

And  Xa/     3a"4-2v/2(a*-f^)»  by  extracting  the  root 
Where,  by  substitutino:  13  for  a,  and  4721  for  6, 
we  shall  have  x=-<i. 

Therefore  — - — = — =8,  the  greater  number, 

,     ,  13— r      10  ^     , 

And  — - — =—:=->   the  less  number, 
2  2 

The  sum  of  which  is  13,  and  8  -{-5*=47?1. 

10.  Given  the  sum  of  two  numbers  equal  5,  and  their 
product  =p,  to  find  the  sum  of  their  squares,  cubes,  bi- 
rjuadrates,  &c. 

Let  x  and  y  denote  the  two  numbers  ;  then 
(I.)  ;r4-2/  =  s,  (f.)  ay=p. 
From  the  square  of  the  tirst  of  these  equa  ions  take  twice 
the  second,  and  we  shall  have 

(8.)  x^-\-y^=s-^--2p=  sum  of  the  squares. 
Multiply  this  by  the    1st  equation,  and  the  product  will  be 
xr-]-xy~-\-x^y-{-y=s^'-'2sp. 

From  which  subtract  the  product  of  the  first  and  second 
equations,  and  there  will  remain 

(4.)  x" 4-^^=5^  — 3.vp=  sum  of  the  cubes. 
Multiply  this  likewise  by  the  1st,  and   the   product  will  be 
x'^-\-xy^'{-x-y-\-y^=s^     Ss"p  ;  from  whicli  subtract  the  pro- 
duct of  the  second   and  third  equations,  and  there  will  re- 
main 


144  QUADRATIC  EQUATIONS. 

^5,)  -p''4-y''-~«'*-  4s^p+2p^-=  sum  of  the  biquadrates* 
And,  again  multiplying  this  by  the   1st  equation  and  sub^ 
tracting  from  the  result  the   product  of  the  second  and 
fburth,  we  shall  have 

(6.)  x^-hy^~s^     biy-\-5sp^=  sum  of  the  fifth  powers. 
And  so  on  ;  the  expression  for  the  sum  of  any   powers  in 

„      ,   mfm  — 3)      .  „ 
general   being   x'^+y  =  s^'^-ms'^'^p -\ ^ s'^-Y" 

7n(m~4)(m-~5)  mjm^b)  (m->6)  (m~7)  _ 

2-3  ^  2-3-4  ^       ■ 

&c. 

Where  it  is  evident  that  the  series  will  terminate  when  the 
index  of  s  becomes  =o. 

EXAMPLES  FOR  PRACTICE. 

1.  It  is  required  to  divide  the  number  40  into  two  such 
parts,  that  the  sum  of  their  squares  shall  be  Si 8. 

Ans.   23  and  17. 

2.  To  find  a  number  such,  that  if  you  subtract  it  from 
1 0,  and  then  multiply  the  remainder  by  the  number  itself, 
the  product  shall  be  21.  Ans.   7  or  3, 

3.  It  is  required  to  divide  the  number  24  into  two  such 
parts,  that  their  product  shall  be  equal  to  '^5  tjmes  their 
difference.  Ans.  10  and  14. 

4.  It  is  required  to  divide  a  line,  of  20  mches  in  length, 
into  two  such  parts  that  the  rectangle  of  the  whole  and  one 
of  the  parts  shall  be  equal  to  the  square  of  the  other. 

Ans.    10^6- 10,  and  30-  10^/5. 

5.  It  is  required  to  divide  the  number  60  into  two  such 
parts  that  their  product  shall  be  to  the  sum  of  their  squares 
5n  the  ratio  of  2  to  5.  Ans.  20  and  40. 

6.  It  is  required  to  divide  the  number  146  into  two  such 
parts,  that  the  difference  of  their  square  roots  shall  be  6. 

Ans.   25  and  12  K 

7.  What  two  numbers  are  those  whose  sum  is  20  and 
their  product  36  ?  Ans.  2  and  18. 

8.  The  sum  of  two  numbers  is  I^,  and  the  sum  of  their 
reciprocals  3^  ;  required  the  numbers.  Ans.  ^  and  |- 


^UABRATIC  EQUATIONS.  145 

9.  The  difference  of  two  numbers  is  15,  and  half  their 
product  is  equal  to  the  cube  of  the  less  number  ;  required 
'he  numbers.  Ans.  3  and  18* 

10.  The  difference  of  two  numbers  is  5,  and  the  differ* 
snce  of  their  cubes  16^5  ;  required  the  numbers. 

Ans.  8  and  18. 

11 .  A  person  bought  cloth  for  33Z.  1 5s.  which  he  sold 
again  at  2/.  8s.  per  piece,  and  gained  by  the  bargain  as 
much  as  one  piece  cost  hsm  ;  required  the  number  of 
pieces.  Ans.  15. 

12.  What  two  numbers  are  those,  whose  sum  multipli- 
ed  by  the  greater,  is  equal  to  77,  and  whose  differencCj 
multiplied  by  the  less,  is  equal  to  12.  Ans.  4  and  7„ 

13.  A  grazier  bought  as  many  sheep  as  cost  him  60Z., 
and  after  reserving  15  out  of  the  number,  sold  the  remain- 
der for  54/.,  and  gained  2s.  a  head  by  them  :  how  many 
^heep  did  he  buy  ?  Ans.  75. 

14.  It  is  required  to  find  two  numbers,  such  that  their 
product  shall  be  equal  to  the  difference  of  their  squares, 
and  the  sum  of  their  squares  equal  to  the  difference  of 
their  cubes.  Ans.  ^^5  and  l{5-{-  ^/  5)-. 

1 5.  The  difference  of  two  numbers  is  8,  and  the  dif- 
ference of  their  fourth  powers  is  14560 ;  required  the 
numbers.*  ^  Ans.  3  and  11, 

*  In  solving  this  question,  the  reduced  equation,  found  by  the  usual  me- 
thods of  operation,  will  be  of  the  form  x3  -f-aa:=  b  ;  which  is  a  cubic  equa- 
tion, and  therefore  cannot  be  resolved  by  the  ordioary  rules  of  quadratics  ; 
but  such  equations  may  sometimes  be  reduced  to  the  form  of  a  quadratic, 
and  then  resolved  according  to  the  rules  already  given. 

Whenever,  in  a  cubic  equation  of  the  form  a?3  -J-  aa;  =sb  ;  b  can  be  divided 
into  two  factors  m  and  n,  so  that  m2  ~^a-=n,  then  the  cubic  equation  can  be 
resolved  as  a  quadratic  :  thus,  in  the  cubic  equation  a;  3  4.  6a;  =20, 20  =  2X 
iO,  and  22  -|-  6  =  10.  Now,  multiplying  both  the  sides  of  the  equation  by  x, 
we  have  a;4  -f-6a:3  =  10  x  2x,  adding  (2a;)2  to  both  sides,  a;4  -f-lOxS  c=» 
(2a;)2  J-10(2a;);  .-.  completing  the  square, 

a  4  -j-  10a:2  -f  25  =  (2x)2  + 1 0(2x)  +  25, 
and  extracting  the  root,  x2  -|-5=  2x4-5  ;  •"•  by  transposition,  xa  s=2x, 
and  x<=  2,  or  a=  0. 

This  method,  as  well  as  some  other  similar  artifices,  is  of  no  utility  whea  the 
divisor  has  not  integral  roots,  and  even  then  it  can  be  resolved  more  readi' 
'1/  by  JVewton's  Method  of  Divisors. 

It  is  proper  to  observe  that  cubic  equations  of  the  form  a;3-J-aa;3-|-6a!  «=  c, 
nmay  be  also  exhibited  under  the  form  of  a  quadratic,  from  which  by  comp'et 
O 


146  QUADRATIC  EQUATIONS. 

16.  A  company  at  a  tavern  had  bl.  \5s.  to  pay  for  theif 
reckoning  ;  but  beloie  the  bill  was  Bettled,  two  of  them 
went  away  ;  lu  consequence  of  vhich  those  who  remained 
had  lO.s.  a  piece  more  to  pay  than  before  ;  how  many  were 
there  in  company  ?  Ans.  7. 

17.  A  person  ordered  71.  45.  to  be  distributed  among 
some  poor  people ;  but  before  the  money  was  divided, 
there  came  in,  unexpecltdlv,  two  claimants  more,  by  which 
means  the  former  received  a  &.hiihng  a  piece  less  than  they 
would  otherwise  have  done  ;  what  was  their  number  at 
first?  Ans.  16  persons. 

18.  It  is  required  to  find  four  numbers  u.  geom.etrical 
progression  t-uch,  that  their  sum  shall  be  i5,  and  the  sum 
of  their  squares  85.  Ans.  1,  2,  4,  and  8* 

"1.  The  sum  ottwo  numbers  is  11,  and  the  sum  of  their 
fifth  powers  is  ;783l  ;  required  the  numbers? 

Ans.  4  and  7* 

20.  It  is  required  to  find  four  numbers  in  arithmetical 
progression  such,  that  their  common  difference  shall  be  4, 
and  their  continued  product  17H98  >. 

Ans.  15,  19,23,  and  27. 

21.  Two  detachments  of  foot  being  ordered  to  a  station 
at  the  distance  of  3:'  miles  from  their  present  quarters, 
begin  their  "march  at  the  sanie  time  ;  but  one  party, 
by  travelling  i  of  a  mile  an  hour  faster  than  the  other, 
arrive  there  an  hour  sooner ;  required  their  rates  of 
marching?  Ans.  3^  and  3  miles  per  hour. 

22.  it  is  required  to  find  two  numbers  such  that  the 
square  of  the  first  plus  their  product,  shall  be  140,  and  the 
square  of  the  second  minus  their  product  78. 

Ans.  7  and  13. 

23.  It  is  required  to  find  two  numbers,  such  that  their 
difference  shall  be  13  -^Ye'  ^^'^  ^''®  difference  of  their  cube 
roots  Ij.  Ans.  I5f,  and  2i-a. 


ing  the  square,  the  value  of  (he  unknown  quantity  will  be  determined.  For  in- 
stance, the  cubic  equation  ac  3  -j-  2:ix2  -^  5a2  jr  i  4fi3  =:  0,  may  be  reduced  to 
the  forra  (x2  -^aa:)2  _{_4ft2(a.'2 -}_ax^i=0;  thus,  muhiply  the  given  equa-~ 
lion  by  *,  we  have  xi  -f-  2ax^  -f-  ^«2ar2  -|-4a3x  =0  ;  whiih  may  be  readi- 
ly exhibited  under  trie  above  form  :  see  Rvan's  Elementary  Treatise  on  Al- 
gebra, Prattieai  and  Theoretical.     'Ait.  423.)  '  V* 


QUADRATIC  EQUATIONS.  147 

24.  It  19  required  to  find  three  numbers  in  arithmetical 
progression,  such  that  the  ^um  of  their  squares  shall  be  93  j 
and  if  the  first  be  multiphed  by  <,  the  second  by  4,  and  the 
third  by  5,  the  sum  of  the  products  shall  be  66. 

Ans.  2,  5,  and  8« 

25.  The  sum  of  three  numbers  in  harmonical  propor- 
tion is  191,  and  the  product  of  the  first  and  third  is  4032  \ 
required  the  numbers.  Ans.  72.  63,  and  56, 

26.  It  is  required  to  find  four  numbers  in  arithmetical 
progression,  such  that  if  they  are  in-reased  by  2,  4,  8.  and 
16  respectively,  the  sums  shall  be  in  seometrical  progres* 
sion.  Ans.  6,  ^,  *0,  and  12. 

27.  It  is  required  to  find  two  numbers,  such,  that  if  their 
diflTerence  bo  multiph-^i  into  their  sum,  the  product  will 
be  3 ;  but  if  the  difference  of  their  squares  be  multiplied 
into  the  sum  of  their  squares  the  product  will  be  65. 

Ans.  8  and  2, 

23.  It  is  required  to  divide  the  number  in  inio  two  such 

parts,  that  if  the  square  root   of  the  irreater  part   be  taken 

from  the  greater  part,  the  remainder   shall   be  equal  to  the 

square  root  of  the  less  part  add^d  to  the  less  part. 

Ans.  .s+i^i9and  5— i^l9. 

29.  It  is  required  to  find  two  numbers,  such  that  if  their 
product  be  added  to  their  sum  it  shall  make  61,  and  if 
their  sum  be  taken  t>om  the  sum  of  their  squares  it  shall 
leave  88.  Ans.  7-h^2  and  7 — ^2, 

30.  It  is  required  to  find  two  numbers,  such  that  their 
difference  multiphed  by  the  'Hfference  of  their  squares 
shall  be  j76,  and  their  sum  multiplied  by  the  sum  of  their 
squares  shall  be  2o3  :.  Ans.  5  and  11. 

31.  It  is  required  to  find  three  nu'nbers  in  continual 
proportion,  whose  sum  shall  be  20,  and  the  sum  of  their 
squares  1 40.  Ans.  6|  -f -v/3/^,  dx,  and  6^  -  ^3/^. 

'2.  It  is  required  to  find  tv\'0  numbers  whose  product 
shall  be  320,  and  the  differen<-e  of  their  cubes  to  the  cube 
of  their  difference,  as  61  is  to  unity.  Ans.  20  and  16, 

33.  The  sum  of  700  dollars  was  divided  among  four 
persons,  a,  b,  c  and  d,  whose  shares  were  in  .geometrical 
progression ;  and  the  difference  between  the  greatest  and 


up  CUBIC  EQUATIONS. 

least,  was  to  the  difference  between  the  two  means,  as  3f 
to  12.     What  were  all  the  several  shares? 

Ans.  108,  144,  192,  and  256  Dollars 

OF  CUBIC  EQUATIONS. 

A  cubic  equation  is  that  in  which  the  unknown  quantity 
rises  to  three  dimensions  ;  and  like  quadratics,  or  those 
of  the  higher  orders,  is  either  simple  or  compound. 

A  simple  or  pure  cubic  equation  is  of  the  form 

b  h 

ax^=b,  or  a^=—  ;  where  x==^l/-, 
a  a 

A  compound  cubic  equation  ia  of  the  form 

x^  —  ax=b,  x^-\'ax^=^h,  or  x^'-\-nx''-\-bx=Cf 
in  each  of  which,   the  knov/n  quantities  a,  b,  c,  may  be  e^ 
ther  -{-  or  —  . 

Or,  either  of  the  two  latter  of  these  equations  may  be 
reduced  to  the  same  form  as  the  first,  by  taking  away  its 
second  term  ;  which  is  done  as  follows  : 

RULE. 

Take  som«  new  unknown  quantity,  and  subjoin  to  it  a 
third  part  of  the  coefficient  of  the  spcond  term  of  the 
equation  with  its  sign  changed  ;  then  if  this  sum,  or  dif- 
ference, as  it  may  happen  to  be,  be  substituted  for  the 
original  unknown  quantity  and  its  powers  in  the  propos- 
ed equation,  there  will  arise  an  equation  wanting  its  se. 
cond  term. 

JVote.  The  second  term  of  any  of  the  higher  orders  oi 
equations  may  also  bo  exterminated  in  a  smiilar  manner, 
by  substituting  for  the  unknown  quantity  some  other  un- 
known quantity,  and  the  4th,  Sth,  &c.  part  of  the  coeffi- 
cient  of  its  second  term,  with  the  sign  changed,  according 
as  the  equation  is  of  the  4ih,  5th,  &c,  power.* 

*  Equations  may  be  transformed  into  a  variety  of  other  new  equations ;  the 
principal  of  which  are  as  follows  : 

1.  Theequation  i4— 4x3— 1912 -|-106a;— 120  =aO,  the  rootsof  which  are 
2,  3,  4,  and  — 5  ;  by  changing  the  signs  of  tlie  second  and  fourtfi  terms,  be 
comes  x4  4.  4x3— 19.Ta— 106x— 120  «  0,  the  vootfi  of  which  are  5,-2,—^ 
anvl  —4. 


eUBIC  EQUATIONS.  149 


EXAMPLES. 

r.  U  is  required  to  exterminate  the  second  term  of  the 
equation  a;^-f-3ax^=Z>,  or  x^-\-Sax^ —b=0. 

Here  a; =2 =z  -  a. 

Then  {  :^ax^=    ^-Sas^-ea^^+Sa^ 
/   -6=  -6 


Whence  2='-3a2r-{-2a^--6=0, 
Or  ^^~3a"^=6 — 2a^, 
in  which  equation  the  second  power  (z^),  of  the  unknown 
quantity,  is  wanting. 

2.    Let  the  equation  x^ — \'ix^-\'3x=^ — 16,    be  trans- 
iormed  into  another,  that  shall  want  the  second  term. 
Here  x=-z-\"^. 

Then^  -.12(^4-4)2=-  122^ -.96^—192 
I     -1-3(24-4)  =  -f  32+12 

Whence  ^^- 452— 11 6= —  16 
Or  2'- 452=  100 
which    is  an  equation  where  2^  or  the  second   term,  is 
wanting,  as  before. 

2.  The  equation  a;3-f-a;2 — lOx -f- 8  =  0,  is  transformed,  by  assurnjnga:s=y 
— 4,  into  2/3 — li?/2  -{-30?/«=0,  or3/2 — 113/4-30=0;  the  roots  of  which  are 
greater  than  those  of  the  former  by  4. 

3.  The  equation  a' 3 — 6x2  ^9a; — 1  =s  0,  may  be  transformed  into  one  which 
shall  want  the  third  term,  by  assuming  x  =  y-^e,  and  in  the  resulting  equa- 
tion, let  3c2 — 12e-f-9,  or  e3— 4e-f-3  =  0,  in  which  the  values  of  e  are  land 
3;  then  assume  x  —  y-v'^,  or  y  +  i,  and  the  resulting  equation  will  beyS-f- 
3y2 — 1  =  0,  an  equation  wanting  the  third  term. 

4.  The   equation  6a; 3— 1U2 -J-.6x—l  =3  0  by  assuming  a;'^ -,  may  be 

transformed  into  2/3 — 6y2  -{-Wy — 8  =0  ;  the  roots  of  which  are  to  be  reci- 
procals of  the  former. 

V 

5.  The  equation  3r3 -13x2  ^-14a; 4-16— 0,  by  assuming  a;s^^,  may  be 

transformed  into  y^ — \3y2  -f.42y  -f  144  =0,  the  roots  of  which  are  three 
limes  those  of  the  former,  Ed. 

o2 


150  CUBIC  EQUATIONS. 

3.  Let  the  equation  x^— 6x^=10,  be  transformed   into 
another  that  shall  want  the  second  term. 

Ans.  if-  122/=26. 

4.  Let  if—]by^'\-81y='zA3,  be  transformed  into  an 
equation  that  shall  want  the  second  term. 

Ans.  «^-|-6a;  =  88. 

3  7         9 

5.  Let  the  equation  x^-\ — x^-\--x =  0,     be    trans- 

4  8        16 

formed  into  another,  that  shall  want  the  second  term. 

Ans.  ?/^+_2/=-. 

6.  Let  tho  equation  x'-\-dx-^ — 5s---{-10j?  — 4  =  0,  he 
transformed  into  another,  that  shall  want  the  second  term. 

Ans.  ?/^  —  '29jy= 4-94?/— 92  =  0. 

7.  Let  the  equation  x^  — 3x^— 3a;^- 5a:  — 2=0,  be  trans- 
formed into  another,  that  shall  want  the  third  term. 

Ans.  y^-\-y^~4y — 2  —  0. 

8.  Let  the  equation  3x^— 2x-i-l  =  0,  be  transformed 
into  another,  whose  roots  are  the  reciprocals  of  the  for- 
mer. Ans.  i/^--2i/'^+3  =  0, 

9.  Let  the  equation  x''  — ^x^+i.r^-fx-fyV— ^^  ^^ 
transformed  into  another,  in  which  the  coefficient  of  tht 
highest  term  shall  be  unity,  and  the  remaining  terms  inte- 
gers, Ans.  y'-~3y^-\'\'2y^—\62y-\-72=0. 

OF  THE  SOLUTION  OF  CUBIC  EQUATIONS. 

RULE. 

Take  away  the  second  term  of  tho  equation  when  ne 
cessary,  as  directed  in  the  preceding  rule.  Then,  if  the 
numeral  coefficients  of  the  given  equation,  or  of  that 
arising  from  the  reduction  above  mentioned,  be  substituted 
for  a  and  b  in  either  of  the  following  formulae,  the  result 
will  give  one  of  the  roots,  as  required*. 

*  If,  instead  of  (he  regulat  method  of  reducing;  a  cubic  equation  of  tht 


CUBIC  EQUATIONS.  151 

or 

^j  2+^(7+27)^ 

Where  it  is  to  b3  observed,  that  when  tl.e  coefficient  Uj 
of  the  second  term   of  the  above  equation,   is   negative. 

— ,  as  also  -,  in  the  formula,  will  be  negative  ;  and  if  the 
absolute  term  6  be  negative,  -  in  the  formula,  will  also  be 
negative  ;   but  ■— -  will  be  positive.* 

a;  3  -f  «^  +  ta;  "4-  c  =  0. 
to  another,  wanting:  the  second  term,  as  pointed  out  in  the  preceding  article, 
there  be  put,  x  *=»  -^(y — a),  we  shall  have,  by  substitution  and  reduction,  y» 
-|-(96— 3a2)2/=  9ab — 27c — 2tx3  ;  where,  since  the  value  of  y  can  be  deteC'^ 
mined,  by  either  of  the  formulae  given  in  diis  rule,  the  value  of  a:  will  also  be 
known,  being  a;  =^^iy — a).  And  if  6=  t),  or  ihe  original  equation  be  of  the 
following  form  a;  3 -f-a.r  a  -^c=  0,  the  rtduced  equation  will  beys— Sazy^— 
— 2a3 — 27c,  where  the  value  of  y,  bi-in^  found  as  above,  we  shall  have,  as 
before,  x=z^{y — o),  which  formulas,  if  may  be  observed,  are  more  convenient, 
in  some  cases,  than  those  resulting  from  the  preceding  article  ;  as  iho  co- 
efficients, thus  obtained,  are  always  in'.egers;  whereas  by  the  former  method, 
they  are  frequently  fractions. 

*  The  method  of  solving  cubic  equations  is  usually  ascribed  to  Cardan, 
a  celebrated  Italian  analyst  of  the  16th  century  ;  but  the  authors  of  it  were 
Scipio  Ferreus,  and  Nicolas  Tartalea,  who  discovered  itibout  the  same  time, 
independently  of  each  other,  as  is  pioved  by  Montucla,  in  his  Histoire  des 
Mathematiqv.es,  Vol.  I.  p.  568,  and  more  at  large  in  llyMori's  Mathematical 
Dictionary,  Art.  Algebra. 

The  rule  above  given,  which  is  similar  to  that  of  Cardan,  may  be  demon* 
itrated  as  follows : 

Let  the  equation,  whose  root  is  required,  be  x3  -|-axc=  l). 
And  assume  y  ■\-z  =sx,  and  3yx  «=  — o. 
Then,  by  substituting  these  values  in  the  given  equation,  we  shall  havfe 
v34-3ya2  4.3y«3  4.2^  +  a  X(y+2)  =y3  -fzs  -\-3yzX  ^V-h  «)+«  X 
^  yJl  r)  =  y  3  4-2 3  — a  X  (2/+^)  +  «  X  (y + 2r)  =a6,  or 

y3-f.«3=x6. 

And  if,  from  the  square  of  this  last  equation,  there  be  taken  4  times  the 
icube  of  the  equation  yz  a — -^cp,  we  shall  have  y6—2y3«3 -J-;??  =3^2  4» 
-i^ns,  or 


152  CUBIC  EQUATIONS. 

It  may  likewise   be  remarked,  that  when  the  equatioj* 
is  of  the  form 

and   —  is  greater  than  -— ,  or  4a »    greater  than   276  2,  the 

solution  of  it  cannot  be  obtained  by  the  above  rule  ;  as 
the  question,  in  this  instance,  falls  under  what  is  usually 
called  the  Irreducible  Case  of  cubic  equations.* 


y3—Z3  =  ^/(63  _f  2^03) 

But  the   sum  of  this  equation  and  1/3    ir3  =  6,  is  2y3  tKb^^(/j2 
■^jaS)  and  their  diiference    ia  2z3  =^b—^{h2  4.  g^aS)  •,  whence  y=  v^ 

From  which  it  appears,  that  y\  z,  or  its  equal  a,  is  = 

^{\b^  V^jbi^ijaz))  ^l/{,^b  —  ^{\b2^^^a^)),  which   is   thf 

theorem  ; 

a    .      .,,  ,  a 

Or,  siuce  *  is  = — — ,  it  will  be  3/4.^=3/— —,  or  a;  r= 

^y  ^  ^y 

V{\b  +  ,/i.\b2  +^\a.))—^^^j^-^^^,  the  same  as  the 

rule. 

*  It  may  here  be  farther  observed  as  a  remarkable  circumstance  in  the  his- 
tory of  this  science,  that  the  solution  of  the  Irreducible  Case  above  mentioned, 
except  by  means  of  a  table  of  sin^s,  or  by  infinite  series,  has  hitherto  baffled 
the  united  eflbrts  of  the  mcjst  celebrated  mathematicians  in  Europe  ;  although 
it  is  well  known  that  all  the  three  roots  of  the  equation  are,  in  this  case,  real  ; 
whereas  in  those  that  are  resolvable  by  the  above  formula,  only  one  of  the 
roots  is  real,  so  that  in  fact,  the  rule  is  only  applicable  to  such  cubics  as  have 
\wo  equal,  or  two  impossible  ro<jts. 

The  reason  why  the  assumptions,  made  in  the  note  to  the  former  part  of 
this  article  with  respect  to  the  solution  of  the  equation  :r3 — ax  =6,  are  founri 
to  fail  in  tiie  case  in  question  (:iiid  it  does  not  appear  that  any  other  can  be 
adopted)  is,  that  the  two  auxiliary  equations  Syz  ^ —a  and  y3-\-z'i  ■=  k. 
which  in  this  case,  become  3yz=:a,  and  3^3  1  2:3— 6,  or  y3z3K= 
a3 
—,  and  2/ 3 -f-^^ '^^j  cannot  take  place  together;  being  inconsistent  with 

each  other. 

For  the  greatest  product  that  can  be  formed  of  the  two  quantities  ys-j-zs 
is,  when  they  are  all  equal  to  each  other  ;  or  since  y3  ^z3'=  b,  when  each 
t>f  these  =r  -g-i  ;  in  which  case  their  product  is    =  jb2  . 

But,  as  above  shown,  3/32  3=  —  ,  by  the  question,  therefore  when  °^>— , 

the  two  conditions  are  incompatible  with  each  other  ;  and  consequently  the 
solution  of  the  problem,  upon  that  supposition,  can  only  be  obtained  by  ima 
ginary  quantities. 


etJBiC  EQUATIONS.  i&3 


EXAMPLES. 

K  Given  2x^ — ISx^-f-jdx— 44,  to  find  the  value  of  tr. 
Here  x'^ — 6x2H-18r  =  :^2,  by  dividing  by  2, 
And,  in  order  to  exterminate  the  second  term. 

Put  x=2+-=2'-|-2, 
o 

(^  +  2)'*:  z-f  6^24-!  2^+8 

Then     — 6(2-4-2)^=    — 6^-^—2  Iz— 24     ==22 

lS(z-f'^)  =  18^4-36 

Whence  z'^A-e^z  -^20=  i 2,  or  z''+6z=2. 
ind  consequently,  by  substituting  6  for  a,  and  2  for  6,  its 
^he  first  formula,  we  dhall  have, 

V(i4-A/(i+B))-i-v'(i-y(i4-s))=y(i4-v/9)4^ 

3/(i_^9)  =  y(l+a)4-V(l  -3)  =  V4- ^2, 
Therefore  2;^z=f  2^2/4— s/"2 4-2=2-1-1.587401 -». 
1.25992 1  =2?.32r 48,  the  answer. 
2.  Given  x^-— 6 1  =12,  to  find  the  value  of  x> 
Hero  a  being  equal  to  —  ^,  and  6  equal  to  12,  we  shall 
have,  by  the  formula, 

V(6W28)  +  37^^^i^  =  V(6+5.29I5)  + 
^  =V(ll.29.5)4-  L^   2  =  .243a+ 


y(6  +  5.2915)     ^'  ''3/(11.^:915) 

2 

=2.2435+, 8957=3. 1392 

2.2435 

Therefore  a;=^^,'392,  the  answer. 
3.  Given  x^  — 2x=— 4,  to  find  the  value  of  x. 
Here  a  being  =-2,  and  6=— 4,  we  shall  have,  by  th» 
formula, 

^=Vf-2+v'(4-|,-)!+Vf-2-v'(4-^)h  n 


ifl4  CUBIC  EQUATIONS. 

Ii7  reductioQ,  i^(— 24-~>/3)-3/(24-~-v/3)= 

V( --'4-1.9245)— 3/(-i.fI.9246)=yr— .0755—= 
^3.9245  -—  1 226—- 1 .  ,77.^=—  1 .99  99,  or  —2 
Therefore  /— - — -',  the  answer* 
Note,  When  one  of  the  rotus  of  a   cuhic  equation  has 
been  found,  by  the  common  tormula  as  above,   or  in  any 
other  way,  the  other  two  roots  may  be  determined  as  fol- 
lows : 

Let  the  known  root  he  do.rmted  by  r,  and  put  all  the 
terms  of  the  equation,  whei  orou<jht  to  the  left  hand  side, 
=0  ;  then  if  the  equfition,  so  forned,  be  divided  by  ar±:r, 
according  as  r  is  posnne  or  negative,  the»e  will  arise  a 
quadratic  equation,  the  roots  of  which  will  be  the  othei: 
two  roots  of  the  given  cubic  equation. 

4.  Given  x^ — 15<  =  4,  to  find  the  three  roots,  or  values 
of  X. 

Here  x  is  readily  found,  by  a  few  trials,  to  be  equal  t« 
4,  and  therefore 

X — 4)a?  —  I5x — 4(a;^+4a?+l 
x-—-\£^ 


4a:2— 15a; 


X— 4 

a— 4 


*  When  the  root  of  the  given  equation  is  a  whole  number,  this  method  onU 
determines  it  by  an  approxirnation  of  9>  m  the  decimal  part,  which  sufli- 
ciently  indicates  the  entire  inte.er  ;  i.iit  in  most  instances  of  this  kind,  its 
value  may  be  more  readily  found,  by  u  .ew  trials^from  the  equation  itself. 

Or  if,  as  in  the  above  exam[}l(^,  the   rents,  or  numeral  values  of  \/( — S-fr 

^v/3),  and  — v^>2-+---  v/3)be  di-termii'ed  accordini^to  the  rule  laid  dovrn 
y  9 

h  Surds,  Case  12,  the  result  will  be  lound  equal  to  —2  as  it  ought-. 


tJUBIC  EQUATIONS.  IBB 

Whence,  according  to  the  note  above  given, 
x^'^'kt-   t— <).  or  j^-l-4x~  —  i  ; 
the  two  roots  of  which  quadratic  are  —  2-ry/3  and  — 2  — 
\/3't  and  consequeiiuy 

4,   -      -t-v/3,  and  —2  -  v/  3, 
are  the  three  roots  of  the  proposed  equation. 
Or,  putting  a--      lo  and  6  =  4,  we  shall  have, 

^(2-hv^  — i.l)-2  +  v'-~^ 

3/(2-v/-l2l)  =  2-v^-l, 
as  will  be  found  either  by  eubmg  .^+^ — 1  and  2  — y'  — l., 
or  by  the  rule  given  in  case  i'-Z  surds. 

W hence  (/-f  c  =  ^2 4- y— « +"2  -  v^— 1  =4, 
-K^+c)-|-i(ri-c)y-3=:  -  2-f-v^-  1  X  v^-3  =  -. 

2-^/3, 
~K^+c)— i(^-c)v/-.-^-=-^-  V  -  1  Xv'-  3=  - 

2  +  v/.3  ; 
and  consequently  4,  — -i — y'  S  and  — 2-}-  \/  3  are  the  thre© 
roots  of  the  equation,  as  before  found. 

EXAMPLF.S  FOR  PRACTICE. 

1.  Given  y^-}-3j^— 6r=8,  to  find  the  root  of  the  equa» 
tion,  or  the  value  of  x.  Ans.  x=^2, 

2.  Given  x^-l-x^=600,  to  find  the  root  of  the  equation, 
cr  the  value  of  X.  Ans.  a:  — 7.6  15789. 

3.  Given   x^~-3a;^— 5,  to    find  the  root  of  the  equatioHj 
or  the  value  of  x.  Ans.  a-  —  :3. 103803. 

4.  Given  a;^— 6a:  =  6,  to  find    the    root  of  the   equatioHj 
or  the  value  of  x.  Ans.  1/4-^^2. 

5.  Given  a;^+9^=6,  to  find  the   root  of  the   equation, 
or  the  value  of  ^r.  Ans.  ^9  — ^3. 

6.  Given  x3+2a;^-23x=70,   to  find   the   root  of   the 
equation,  or  the  value  of  x.  Ans.  a;= 5. 134899, 

7.  Given  a:=^—17u;"+ 54x^^360,  to  find   the  root  of  thr 
equation,  or  the  value  of  x.      *        Ans.  a;=  14.964068. 


1B6  CUBIC  EQUATIONS. 

8.  Given  a;^— ea:=4,  to  find  the  three  roots  of  Uie 
e^uatioo,  or  the  three  values  of  x. 

AfiH.  —  2,  l+-y/3,  andl  — ^3. 

9.  Given  x^ — 6x--f2a;= — 12,  to  find  the  three  roots  of 
the  equation,  or  the  three  values  of  x. 

Ans.  — 3,  i+y^6,  and  1  —  ^6. 

OF  THE 

SOLUTION  OF  CUBIC  EQUATIONS, 

BY 

CONVERGING  SERIES. 

This  method,  which,  in  some  cases,  will  be  found 
more  convenient  in  practice  than  the  former,  consists 
in  substituting  the  numeral  parts  of  the  given  equation,  in 
Ihe  place  of  the  literal,  in  one  of  the  following  general 
formulae,  according  to  which  it  may  be  found  to  belong, 
and  then  collecting  as  many  terms  of  the  series  as  are  suf- 
ficient for  determining  the  value  of  the  unknown  quantity, 
to  the  degree  of  exactness  required.* 


*  The  method  laid  down  in  this  article,  of  solving  cubic  equations  bv 
means  of  series,  was  first  f;iven  by  Nicole,  in  tlie  Memoirs  of  the  Academy 
of  Sciences,  an.  1738,  p.  99  ;  and  afterwards  at  greater  length,  by  Clairaijt 
in  his  Elemens  dfAlgebre. 

f  Witti  respect  lo  the  determination  of  the  roots  of  cubic  equations  by 
means  of  series,  let  tnere  be  given,  as  above,  the  equation  x  3  _j.ax«=:  6,  where 
the  root  by  transposing  the  terms  of  each  of  the  two  branches  of  the  common 
rbrraula,  is 

2  A  ^    ;   or,  by  putting,  for  the  sake  of  greater  simplicity,  v/(l^2  +  2Va2  ^ 
^=«,  and  reducing  the  expression,  x=as^  \\/  ^1_{.^_)_  iy(i_l)| 


CUBIC  EQUATIONS,  157 

26 i    J-lir     ^^^^  2.5.8.11 

\276=^+4aV  ^6.9.12.  !5.18.2lV2762+4a2y^  ^  S       ' 

26  S  ,  .  2.5,     276^       ,     .   8.11 


i        2.5       276^ 


3/(2(27624-4«^)  <      •  6.9^2762+4a'^     '  12.15 
/     276^      s         1 4. 17/    _276^     V     ,20.^3 

V276^+4tt.V  ^"^12.21  \>7"6H-4aV  ^"^24.27 
276^ 

In  which  case,  as  well  as  in  all  the  following  ones,  a,  b, 
J,  &c.  denote  the  terms  immediately  preceding  those  in 
which  they  are  first  found. 

2.  x^  —  ax=^jr_b,  where  |6^  is    supposed   to  be  greater 
ih&n  ^'^a"",  or  2W>  ia\ 
._^.Q3>  S 2    276^-~4a^         2.5.8      2762--4a3.. 

2.5.8.11.14     ,2762-4a3 


3.6.9,12.15.18'     276 


(^i^V--!-o. 


Hence,  extracting  the  roots  of  the  right  hand  member  of  this  equation,  I  ^ 
.he  binomial  theorem,  there   wiil    arise,  a/i  1-4- -r- )  =j1H-t  (  — ^ — — 
/  6  \      ,    2.5   -r6  X  2.5.8^6-.,^ 

:,/  6>._       j^/6^N 2r6>.^         2     5^6x    _      2.5.S 

f-^)4-&c. 

And  consequently,  if  the  latter  of  these  two  series  be  taken  from  the  former, 
'he  result,  bv  making  the  first  term  of  the  remainder,  a  multiplier,  will  give; 

65     i    ^6  9^2sJ    ^'  '^    S 
vhere,  since  5=  ^/(i  6^  -|-2T«^)»  ^e  shall  have  (^)2  ^  __^    2     ^ 

fb_Y       r-— -^— "^-2    &        A  d  — i~  — -= 26  "      ^ 

•2»>'    '^V2762-j-4a3    *   '    '^'        "      65    ~6.v|  "    ^/(2(«762 -f.4aa)) 

Whence,  also,  by  substitution  we  have  the  above  formula. 

*  The  root,  as  found  by  the  common  formula,  when  properly  reduced,  is 


158  CUBIC  EQUATIONS. 

11.14  2762-4a\        17.20  276'- 4a\ 
-1-7-8^-^^- -^'-2lTr4^-i76->-^^- 

In  which  case  the  upper  sign  must  be  taken  when  b  is 
positive,  and  the  under  sign  when  it  is  negative;  and  the 
same  »'or  the  first  root  m  the  two  following  cases, 

3.   rr^-aj.=+6, 
where  ^6^  is   supposed   to  be   less   than  ^\a^  or  21b~< 
4a=^. 


b  i      ,2     4rr--276\         2. 


2.5.8    ^4a^-276%3 


.9.12^     27^2 

2.5.8.11.14      40=^-27^^3  '   c        ^=.0 

• ^ V— &r.  /  ^.   Or. 

3.6.9.12.15.18^     2ltr      ^       ^^'  ^    -   ^f> 


a-)U  .     Or,  putting,  as  in  the  last  case  -4^/  (^i- ~  2  i a^  \  ov  its  equa": 

Whence,  extracting  the  roots  of  the  right  hand  member  of  this  equaticn.. 

there  will  arise  ^(14-^^)  -^^-^h-^'  +3^^ '"sIlTe^*    +  '^^ 
...  ,      1         2    o       2.5    3       2  5  8.      4       „ 

And,  consequently,  by  adding  the  two  series  together,  and  faking  the  first 

3  -6  t  2    o 

term  of  the  result  as  a  multiplier,  we  shall  have  x=  •±_^'V  -^  \  1 — —.s"  — 

2.5.8      4       2.5. 8.11. U      e      ,,      ?  ^     .         .    ,.'    .      /27i3-4a3x 
3A9.l/'^^:i2l^§'   ~&:c.5  0r,.ysubstuut..g(-^^— J   for 
its  equa!  J,  we  get  the  alx)ve  expression. 


*  This  expression  is  obtained  from  the  last  s^eries,  by  barely  changing  the 
signs  of  the  numerator  and  denominator  in  each  of  its  terms ;  which  does  not 
alter  iheir  value. 

Hence,  in  order  to  determine  the  other  (wo  roots  of  the  equation,  let  (hat 

above  found,  or  its  equivalent  expression  ^  il^+VTi^^ — zV*^  )\ 
Then,  according  (o  the  formula  that  has  been  before  given  '.ot  these  roots. 


CUBIC  EQUATIONS.  169 

,   11.14/4a'-276\        17.20 /4a^- 276% 

which  series  answers  to  the  irreducible  case,  and   must  be 
used  when  -a'  is  less  than  276^ 

And  if  the   root  thus  found    be  put  =r,  the   other   two 
roots  may  be  expressed  as  follows  : 

^_r     ^(4a-  -27fc^)  i         .;.5 /4a'-276\      2.5.  8.11 
X       l-^i-       93/26-'         ^     "6.9^      .-76^      /"^e. 9. 12.15 

\     2762      /       3.6.9.  i2.15.rs.21\       276=^      /^       'V 


in  the  former  pari  of  the  present  article,  we  shall  have  a?  p=  h h  — 

f\/(i6+v^a62— 2V«3))-y(i-6-^(J6-2V^3))|  .     Or,  put- 
''"gyVf  S"^2 — o-V«3   )  =s,  and   reducing   the   expression,   x^^:JP.-~i~ 

*   ^y~-  I  V(t-t-s)— >/(!  —  ?;  {  •     Wlience,  extracting  tliC  cube  roots 
'f  the  right  hand  member  of  ti  is  equation,  tlure  will  arise 

And,  consequently,  by  takinjc  the  latter  of  tnese  series  from  the  former,  and 
making  the  first  term  of  the  remainder  a  mulfiphcr,  we  shall  have  .r  — .qz 
^  j.«(2t)i\^— 3  V,^  2.5  3    .     2581l'       ,     2.5  8.11.1-117   6  } 

2-- 3 V-^Gl^'  -TyJ^Tr  +6.iu2iri-872r  +^'^•1 

I3,jt.; 2   , ,,  ^j,       ,      3^       ,,27/.2~4a3  ,i     2       2762— 4a 3 


6.'.'.12.i-5.18.21 
^    ^/i«.2        I      3v        /ii7/.2 — 4a3     1      2 


£762 
4o3— 2762  x4a3— 2762x3    .  ,    ,       n/i,     «      ,  3/, 


V      «763 
yV— 3   ,/2?62— 4a3x        13  ,  .4a3— 2762x 

><-T-^C-276-2  — J^^^/^^Xv'C-Tji^— )  = 


v/(4a3— 2762) 

9  3  yoA o »  "  *°^S6  values  be  substituted  for  their  equals,  in  the  last  se- 
ries, the  result  will  give  the  above  expressions,  for  the  two  remaining  roots  of 
the  equation. 


460  CUBIC  EQUATIONS' 

__r>s/ (40^-276")  i        2.5 /4a^-27^/~\  S.K 

'^""•+'2—  93/26^  €  '■6.9V~2762  /  ^'^T2:U 
/4a'-27^\  l4.17/4o'^-276^  20.23/4a^~276-v 
\"~276^      /^     18.21  \     27^-      /^'^24r27\     2l¥~ } 

B  — &C. 

"Where— ^r,  or+^r,  must  be  taken  acording  as  o  it 
positive  or  negative ;  and  the  double  signs  j+  must  be 
considered  as  -j-  in  one  case,  and  —  in  the  other,  as  usual, 

4.  x^^ — ax—  ■•  6, 
where  ^h^  is  still  supposed  to   be  less  than  j\a\  or  276- 
^4a^ 

26     i         2.5/     276-      V    , 

^""        vW*«'— 276'J)  ^  ^'"6'.9\4^^'276V'^ 
2.5.8.11/      276^     y      2.5.8.11.14.17/     276^       y 
379712. 1 5  \4a^— 276V       6.9.12. 1 57l8.21  \4a^-.6762/ 

+&C.  I  .     *0r, 


*  By  transposing  the  terms  of  the  conriinon  formula,  as  in  tlic  first  case,  w 
.Vaa^lhavcx^  V  |  V(i62-oVa3)  +  ^6  ^  -^  |^  (l62_^i^„3j  ^ 

36?.     Or,  by  putting,  for  the  sake  of  simplicity,  as  before,  ^Z (\f>^  "iV 

)^  j  =r5,  and  rtducingthe  equation  x  =>x/s  \\/(^l-\-  -.\—^(\ — —  w 

■Whence,  extracting  the  roots  of  the  right  hand  member,  as  in  the  forme 
instances, 

i/(i+i;)"^  +  3  2i)-37gV2^;  +5X9(2^)  -3:6:912(2".)  +'^' 

t//.       ^\       1       i(^\        ^/^\2        2.5,6.3       2  5.8    ,6x4      . 

And  conseqeently,  by  (akng  the  latter  of  these  series  from  the  former. 
sad  making  the  first  term  of  the  result  a  multiplier,  we  shall  have 
.      2^^?  S,.  2.5    6.2  •■5.811      6    4         25  8.11.t4.t7.6.e    . 

■^~    6s     (   "'"6.9X25^    ~ 6.9.12.15  V 2s''     ~6.9.12.15.l8.2r2s'' 

*c.  {  ,    But  since  «=  V  (i^  ^  — 2T'^^\  ^e  shall  have  (—j^  «=."•- 


CUBIC  EQUATIONS.  161 

^~'*'l/1^4aF^iWj)l    ^      6.9^   4a'     ^76^  '^  "*"  12.16 

27P  14.17       276^  ^'0.23       276^ 

Ma^-'276^^^"l87FT^4a^l2762^^"^24.27Mtf^~2762^ 
3— &c.,  which  series  also  answers  to  the  irreducible  case^ 


4a3— 2762\2s>'         V4a3— sr^z^  '  65        6^1 

26 26_     __  2 

*i/(|^2-_.^a3")  3/ J  2(40  3-27^2)  j" 

Whence,  if  these  values  be  substituted  inr  ihf'ir  equals  in  the  last  series, 
here  will  arise  the  above  expression  for  the  first  root  of  the  equation.  And> 
f  we  put  the  root  thus  found,  or  its.  rquivaient  expression 

ve  shall  have,  according-  to  the  formula  before  t^iven  for  the  otlier  two  roots, 

-2^)1  •     0^1  taking,  as  before,  ^ (\^^—2lf>^ )^=  s,  and  simplifying  the 

Whence,  by  extracting  the  roots  o(  the  right  hand  side  of  this  equation, 
liere  will  arise 

'^^'^^s'      ^^•■'\2s)     3.6^2s)    +3.6.94J       3.6.9.12^5''    ^ 
.■J  A,       ^N_i     l/'^^       2/6  y,       2.5/ 6  Ng        2.5.8 

\2s/ 

And,  consequently,  if  the  latter  of  tiiese  series  be  added  to  the  former,  wc 
liall  have,  by  making- the  first  term  of  the  result  a  multiplier, 

^2—       "         i       a6\is/       3.6  9.riV2s/       3.6,8.12.15.18 

[if-^c-    But  since  .^^(ib,-^W>=  (?^;^)iwe.ha!l 

ISO  have  ^^  ;    =  27k^i^  =  -4^^2761*"=-  '""^  '"-^l-™"/.  »5 

Hence,  if  these  values  be  substituted  for  their  equals  in  the  above  series, 
le  result  will  give  the  above  expressions  for  the  two  remaining  roots  of  the 
quatioc 

p  2 


im.  CUBIC  EQUATIONS, 

and  must  be  used  when  2a^  is  greater  than  27b".     And  a 
^he  root  thus  found,  be  put  =r,  as  before,  the  other  two 

loots  may  be  expressed  thus  :  a:=-f--x%/ -r J  l-r' 

2         276^  2.5.8  27i3  2.5.8.11.14 

^^4rt3_-2763^~~3.6.9.T2Ma3-.2762^^  "^.3.6.9.12.15.18 

__r  4a3— 2763  J      ,     2  ^      276^       ,  5.8 

"^-  +  2— ^         4  I  ^'^3:6W-2762/^-~  9.12 

2762  ,11-^4        2762  17.20        276^ 

'Ma3~2762^*^^15^Ma3~'276^^*^~2r.'^Ma3_27p' 

Where  the  signs  are  to  be  taken  as  in  the   latter  part 
the  preceding  case. 

EXAMPLES. 

•     Given  a;"-|-6a;=2,  to  find  the  value  of  x. 
Here  a =6,  and  6=2,  whence 
2762  27X4  f         i 


2762-f4a^     27X4+4X216      1  +  8     9' 

26  _  __    4 

y(2(2762+4«='))  ~^2(4X  27 +4  X'2T6)') " 

4 4    _23/81_y64S  . 

'i'v'{''^^^+8X27)~6379        27       '"~2^~'     ^^^'^^^"^"^  >' 
•>v  formula  1,  we  shall  have 

1  1.0000000  (a) 

gx^A  .0205761  (b; 

S  11       1 

X-F  .0011177(0) 


i2.15     9 

•iljxic  .00007S2(o;. 

''0  28      1 

Ti-«-X-r  .0000062  (e  > 

'24.27     9  ^ 


CUBIC  EQUATIONS.  i65 

§|xJe  .G000005(p) 


1.0217787 


Log.  1.0217787  0.0093570 

Log.  3/648  0.9571916 

Colog.  27  8.5686362 

No.  3274801  —1.5161848 


Therefore  a;=    32748U1 

i.  Given  x^ — 9j:=12  to  find  the  real  value  of  x. 

Here  a~9  and  6=12  ; 

X  o./^2     ^,,^       ,276^-4«='     27X114-4X272 

whence  2V~^V^  and  ._^-^=-— ^^_-_ 

1 44-- 108      36       I 


144  144     4 

^onsequentlv  by  formula  2  we  shall  have 

"i  1.0000000  (a) 

-3|xi(A)  -.0277778  (b; 

^S^l^"'^  -.0025720(0) 

j^^xi(c)  ~- . 000.3667  (d; 

17.20     1  , 

"•'27:24 ''4^-"-^  -.0000619  (E) 

—jXjCf.)  -.00001 14  (r) 
29  32      I 

-33736X4  (^)  -.0000022  (G) 


Sum  --.0307920 


Comp.  .9692080 


u4 


CUBIC  EQUATIONS. 


Log.  969208 

Log.  2^6  or  Log.  ^48 


-1.9864137 
0.5604137 


No.  3.522334  .5478274 

therefore  .t —3.622334. 
3.  Given  x''— 12.t=  16,  to  Rivl  three  values  of  a:. 
Here  a  =  lA  and  6^15  ; 

whence  2y-=2y'-  ^  ^0  and    "  ~ 


2        *-   2 
4.12^—27.152    25m -225      3) 

225 


^z7b' 


27. 152  2i5 

Consequently,  by  formula  'if  we  shall  have 

+  1.0000000  (a; 

4-0.0153086  (b; 
-0.0007812  (c;. 
4-0.0000614  (d) 
—  0.0000067  (e 
fO.0000006  (f; 


_2     3| 

5.   8     31 

X B 

9.12     225 
,   11-14     31 

^15.18     225*" 

17.20     31 
"~2lT2"4^225  ^ 
,  23.26     31 

"^27.30^225  ^ 


Sum  of  +  Terms 
Sum  of —  Terms 

Difference 

Log.  1,0145837 
Log.  \/60 

No.  3.971962 


4-1.0153706 

-  .0007869 


1.0145837 

.0062880 
.5927171 

.5990051 


Therefore  the  affirmative  value  of  x  or  first  root,  rt:^ 
2.9171962. 


CUBIC  EQUATIONS. 


Ui 


"  ^      '        93/26^  93/450       9^^450       3^450 


and 


4a3— 2762      31 


276^ 


225 


+  1 

6  9^225^ 
8  II       31 

14.17      3l_ 

"■lb.2r^225^ 
.  20.23      31 

^24727  ^225 '^ 
26.29      3 1 

■~  30.33  "^  225^ 

Sum 

Log.  97606S3 
Log.  V  93 
Colog.  3/450 
Colog.  3 

No.  .4099445 


Also-- 

Last  No. 

Result 
Or 


Hence. 


1 .0000000  (a) 
-.0255144  (b) 

H-.0017186  (c) 

— 0001490  (d) 

+  .0000145  (e) 

-.0000014  (r^ 

.9760683 


'- 1.9894802 
0.9842415 
9. 11 569. )8 
9.5228787 

-1.6121   64 

-1.9859810 
±0.4099  445 

^1.5760365 
-2.3959255 


Whence  the  three  roots   or  values  of  a-  are   3.971962.. 
—1.5760365,  and  -2.395925. 
.   4.  Given  x^ — 6.i:=2  to  find  the  three  values  of  x. 
-26  -4 

Hprp -r- ,     .TS 

^(2(4a3-  276^))     ^(2(4.6^-27.4)) 


166  CUBIC  EQUATIONS. 

-4 —  4  _  -^  _^   -v/49  27&^ 

3^(2(4.8-4))  ""eyT-S^/?-      ~2T~*  4"a^276  = 
4.27       _     1         1 

'^4.6^-27r4"^8'-rT~'  r 

Hence  by  the  tormula  4,  we  shall  have 

1  1.0000000  (a) 

2.5      1 
_         X^  A  -.0264550  (b) 

"^^^^^  +.0018470(0) 

"l^iiT^J^    '  -.0001662(D) 

20  23     1 
+25i27^7°  +.0000168  (E) 

26.29     1 

-30:^X7^  -.00000.8  w 


Sum-j-. 9752414 


Log.  .9752414  —1.   891120 

Log.  2  0.3010300 

L.  1/49  0.563.^987 

Colog.  21  8.6777807 


No.  3398r0  —  1.5313214 


Therefore  one  of  the  negative  roots  or  values  of  x,  'a- 
^—  .339870=— r. 

.    „    4a3— 2762          4.6^—27.4 
Again  y ^ =%/ -■ =6/(6=^-27)  =V 

J      2762  1 

^«^'^"'' 4-0^27^=7- 

Hence, 
1  1.0000000  (a) 

4-^xiA  +0.0158730(8} 

O.D        7 


5.  8     1 
,  11.14      i 

17.20     1 

"■27724^7^ 

23.26     1 
■^27700^7 ''• 


CUBIC  EQUATIONS.  167 

.0008398  (c) 
-l-.0000b-84  (d) 
-.0000066  (e) 
+  .0000002  (f) 


Sum  1.U150952 


Log.   1.0150952 
Log.  i^l89 

No.  2.431741 


Therefore 


2 


Last  number 

Result 

Or 
And  consequently  -1-2.601676, 
are  the  three  roots  required. 


.006dO7O 
..^794103 

.3859173 

+  .169935 
±2.431741 


+2.601676 

-2.261806 

2.261806,  and— .339870, 


EXAMPLES  FOR   PRACTICE. 

1.  Given  a "+ 9a; =30,  to  find  the  root   of  the  equation, 
or  the  value  of  x.  Ans.  x—2. 1 8C849. 

2.  Given  x^ — 22?=5,  to   find  the  root  of  the   equation, 
or  the  value  of  x.  Ans.  ,r=2. 0945616, 

3.  Given  a:"' — 3a:— 3,  to  find   the  root  of  the  equation, 
or  the  value  of  x.  Ans.  2. 103803. 

4.  Given  x^ — 27a:— 36,  to  find   the   three  roots  or  va- 
lues of  X.      Ans.  5.765722,-4.320684,  and  —1.446038. 

5.  Given  a;^— 48a?-=  — 200,  to  find  the  root  of  the  equa- 
tion, or  the  value  of  x.  Ans.  47.9128. 

6.  Given  x^ — 22a:— 24,  to  find  the  root  of  the  equation, 
or  the  value  of  .t.  Ans.  5,162277. 


168  BIQUADRATIC  EQUATIONS. 

OF  BIQUADRATIC  EQUATIONS. 

A  biquadratic  equation,  as  before  observed,  is  one  thai 
tises  to  the  fourth  power,  or  which  is  of  the  general  fornni 
X  -\-ax  -\-bx'~'{'CX'\'d=^0. 

The  root  of  which  may  be  determined  by  means  of  the 
tbllowing  formula  ;  substituting  the  numbers  of  the  given 
equation,  with  their  proper  signs,  in  the  places  of  the  li- 
teral coefficients  a^  b,  c,  d, 

RULE  I.* 

Pind  the  value  of -jr  in  the  cubic  ecjuation  z^-j-f-ac-^ 

}.b^^dy^^P+l{c'+da^)-^{ac+8d)  by  one  of  the 

former  rules  ;  and  let  the  root,  thus  determined,  be  de- 
noted by  r.  Then  find  the  two  values  ot  x  in  each  of  the 
following  quadratic  equations. 


*  This  method  is  that  given  by  Simpson,  p.  120  of  his  Algebra,  which 
jonsists  in  supposmg  the  given  biquadratic  to  be  formed  by  taking  the  diffe- 
rence of  two  coniplete  squares,  being  the  same  in  principle  as  that  of  Fer- 
-ari. 

Thus,  let  the  proposed  equation  be  of  the  form  i4  'j'ax3~^bx2-{-cx-{'d 
•=0(1),  having  all  its  terms  complete;  and  assume   (jx2  ^■j^ax'\-p)2 — qx 

Then,  if  ar2-}-^aa;-f  p  ar.J    qx^r  be  actually  involved,  we   shall  have 

+  da3 

q2  —2q 


^"^rll    =^'»  +  «^^+*a;2^-ca..f  <f. 


\nd,  consequently,  by  equating  the  homologous  terms,  there  will  arise 


1.  2p-{.\a2—q2  =b 
ap — 2or  =  c 
3.  p:—r2         '    =  d 


2/»  +  |a2— 6=92 
ap  — c  =  2gr 

p2 — d  =  r2 


^vhere,  since  the  product  of  the  first  and  last  of  the  absolute  terms  of  these 
;quations  is  evidently  equal  to  5:  of  the  square  of  the  second,  we  shall  have 

2/>3-{-(ia2 — b)p2 — 2dp — dC^as  — b)=^(a2p2 — 2acp-\-c2). 
Or,  by  bringing  the  unknown  quantities  to  the  left  hand  side,  and  the  knovrc 
'^  the  right,  and  then  dividing  by  2. 


BIQUADRATIC  EQUATIONS.  169 

-"+(J«+V  j  l^'+Kr-lb)  I  )x=-(r+j6)+^  J  (r+ 

and  they  will  be  the  four  roots  of  the  biquadratic  required* 

EXAMPLES. 

1.  Given  the  equation  x*— lOx^+SSa:^— 69a:+24=0,  to 
find  its  roots. 

Here  a— — 10,  6=35,  c= — 50,  and  rf=24  ; 
Whence,  by  substituting  these  numbers  in  the  cubic  equa- 
tion, 


p3—^pi  -\.\(ac—4d)p  =^l(c2  ^a2d)—^bd(2) 

From  which  last  equation  p  can  be  determined  by  the  rules  before  given 
tor  cubics. 

And  since,  from  the  preceding  equations,  it  appears  that 

q=  y/(2p  -l-^a2— 6)andr'=  -^ — ,  or  \/(|J2— c?), 
2q 
it  is  evident  that  the  several  values  of  x  can  be  obtained  from  the  quantities 
ihus  found. 

For,  because  x*-\-ax'i-^bx2  ^cx-^d,  or  its  equal  (a2_4_iaa7-j-p)2 — 
'qx-\-r)2  =0,  it  is  plain  that,(^'2  -f  jac-^p)a  «=  (gx-f-r)2  .     And,  there- 
fore, by  extractincf  the  roots  of  ea<  h  sidt-  of  this  equation,  there  will  arise 
x2  ^  ^ax  +p  =qx  -\-r  \  or  ajs  -^  (-ka — q)x  =ar — p. 
Whence,  by  subsciluting  the  above  values  of  p,  9,  and  r,  for  their  equals, 

and  transposing  the  terms,  we  shall  have  x2  -j-    \^<^^ ^(2p-^jaa — 6)  J 

V  JLpZ^  '^ (p2 — rf)c=  0,  for  the  case  where  ap — c  is  positive  ,•  and 

for  the  case  where  ap — c  is  negative  ;  which  two  quadratics  give  the  four 
roots  of  the  proposed  equation. 

And  by  putting  p  =  «-f--.  in  the  reducing  ev^uation  <2),  in  order  to  destroy 

(s  second  term,  the  several  steps  of  the  investigation  may  be  made  to  a;;ree 
viih  the  expressions  given  in  the  above  rule. 


170  BIQUADRATIC  EQUATIONS. 

we  shall  have  the  following  reduced  equation, 

I'/     108' 
which  being   resolved,   according   to   the  rule  before   laid 
down  for  that  purpose,  gives 

z=}-  ^  3/^33+.8v'— 3)-f-y(35— 18^— 3)  |  . 

But,  by  the  rule  for  binomial  surds,  given  in  the  former 
part  of  the  work, 

^(36-4-18^^— 3)=^4-^v/-3,  and  3/(35-18v'-3) 
=7-1^-3; 

Wherefore.=l  \  ^+l^-3+!-i^--3  \  ^l. 

And  if  this  number  be  substituted  for  r,  — 10  for  a,  35  for  6, 
and  24  for  «/,  in  the  two  quadratic  equations, 

'^+(|«+v/  \  J^'+SCr-^*)  \  )x=.-(r+i6)4-V  \  (r  + 

they  will  become,   after  reducing  them  to  their  most  sim- 
ple terms, 

ics— 3r=^2,  and  :r2-7a=— 12  : 

from  the  first  of  which  x= -^iv^-=-±-=2,  or  1,   and 

7  17     1 

from  the  second  x=-+v/-=-+-=4  or  3; 
2-^4     2—2 

Whence  the  four  roots  of  the  given  equation    are  1,  2,  c), 
>!ind4. 

Or,  when  its  second  term  is   taken  away,   it  will  be  of 


BIQUADRATIC  EQUATIONS.  171 

he  form  x'+br^'+cx+nl^Oy 

0  which  it  can  always  be   reduced  ;  and   in  that  case,  its 

solution  may  be  obtained  by  the  following  rule : 

RULE  II. 

Find  the  value  of  z  in  the  cubic  equation 

md  let  the  root  thus  detf^rmined  be  denoted  by  r. 

Then  find  the  two  values  of  or,  in  each  of  the  following 
[quadratic  equations, 

and  they  will  be  the  four  roots  of  the  biquadratic  equation 
required*. 


*  The  method  of  solving  biquadiatic  equations  was  firsi  discovered  by 
Louis  Ferrari,  a  disciple  of  the  celebrated  Cardan,  before  irentioned  ;  but 
the  nbove  rule  is  derived  from  that  given  by  DesiK  artes  in  his  Geometry,  pub- 
lish'd  in  1637,  the  truth  of  which  may  be  shown  as  follows : 
Let  the  given  or  proposed  equation  be 

x^\-ax^  -j-bx  +  c  =iO, 
and  conceive  it  to  be  produc^^ri  by  the  multiplication  of  the  two  quadratics 
x2i-px-[-q^=0,iind  x'i   ^rx-i'S'=-0 
Then,  since  these  equations,  as  well  as  the  given  one.  are  each  csO,  there 
will  arise,  by  taking  their  product, 

i4-f(p4.r)x3 -4-(s-r94-pr)a;2 -f-(ps  *  qr)x -\- qs  =:  x 4  -f-aa;2  -f-fcx-^-c. 
And  consequently,  bv  f-qnating  the  homologous  terms  of  this  last  equation, 
'vc  shall  have  the  four  followins:  equations, 

p-f-r=0;  s  4~q-{-pr  ^a  ;  ps-^qr:=ib  ;  gs  =  c  \ 

b 
Or,  r  —  —p  ;  «  -r^  y  =  a^pS  ,  5 — q  =  - ,  qs"^  c. 

Whence,  subtracting  the  square  of  the  third  of  these  from  that  of  the  se- 
cond, and  then  changing  the  sides  of  the  equation,  we  shall  have 

a2  -+-i2i;>3  -f  p4 ^qs,  or  4c  ;  or  p 6  -[.2ap4-^-(a2  —Ac)p3 ^^ba . 

p2 
Where  the  value  of  p  may  be.  found  by  the  rule  before  given  for  cubic  equa- 
tions. 

Hence,  also,  since  5-J-}s=a-{-p3 ,  and  s— y=-,  there  will  arise,  by  addi- 

*ion  and  subtraction, 


172  BIQUADRATIC  EQUATIONS. 

Of  the  four  roots  of  the  given  equation,  in  this  last  casCr 
will  be  as  follows  : 

2.  Givt;n  a:*4-l2x  —  i7=u,  to  iind  the  four  roots  of 
the  equation. 

Here  a  =  0,  6  —  0,  <.  =  i2,  and  ^V  =  -.  ]7  ; 

Whence,  by  substituting  these  numbers  in  the  cubic 
equation, 

M2  ^        108    ^8        3     ' 

we  shall  have,  after  simplifying  the  results, 

2'^-[-.7z=   8, 
Where  it  is  evident,  by  nispection,  that  2- —  1. 

And  if  this  number  be  ^libslittitf-d  for  r,  0  for  b,  and 
—  17  for  f/  in  the  two  quadratic  equations  in  the  above 
rule,  their  solution  will  give 

Which  are  the    four  roofs   of  the    propos^ed  equation;  the 
first  two  being  real,  and  the  last  two  imaginary. 


1.1  6     1        I  b 

where  p  being-  known,  8  and  q  are  likewise  known. 

And.  consequently,  by  extraclin^  (he  roots  of  thf  two  assumed  quadratic? 
x2  -J-px-4-9r=0,  and  x2 -f- rx-|-s=0,  or  its  equal  x—px-i~  s=0,  we  shall 
have 

which  expre«siwri,  when  taken  in  -f,  and  — ,  give  the  four  roots  of  the  pre 
posed  biquadratic  as  was  required. 


BIQUADRATIC  EQUATIONS.  173 

RULE*  III. 

The  roots  of  any  biquadratic  equation  of  the  forms  x* 
+aa;^+6x+c=o,  may  also  be  determined  by  the  following 
general  formulae  first  given  by  Eulur  ;  which  are  remark- 
able  for  their  elegance  and  simplicity. 


*  This  method,  which  differs  considerably  from  either  of  the  former,  con- 
sists in  supposing  the  root  of  the  given  equation, 

a:4 -{-ax2  ^6x4-c  =  0  (1), 
to  be  of  the  following  trinomial  surd  form  ;     ^ 

where  p,  q,  r,  denote  the  roots  of  the  cubic  equatioti,  Jff 

y^-hfys  -xgy^h  (2), 
of  which  the  coefficients/,  g-,  and  the  absolute  term  h,  are  the  unknown 
quantities  that  are  to  be  determined. 

Then,  agreeably  to  the  theory  of  equations  before  given>  we  shall  have  p 
-f"? +  »'=—/;  p7-f-pr-f  yr  =  g-;  pqr  =:  h.  And  by  squaring  each  side  of 
the  formula  expressiritc  the  value  of  x, 

x2  ^pJ^q^r  +  H^pq  4  ^^pr-^^s/qr. 
Or,  by  substitutini^  f  iov  its  equal  — (?  +  ?+''),  and  bringing  the  term, 
so  obtained,  to  the  other  side  of  the  equation 

^3  +/=  '2^/pq  +  2^/  pr  +  2v'  qr. 
Also,  by  again  squaring  each  side  of  this  last  expression,  we  shall  have  a* 
-|-2/x2  -j-/2  -=\pq-\-^pr-\-Aqr-j-%^p2qr-^%y/q2pr'\-%*/r3pq. 

Or  substituting  4g-  for  its  equal  4pg+4/>r-j-4gr,  and  bringing  the  term  to 
the  other  side  as  be'bre, 

X 4  +2/C2  4-/2  -4g-  =  8  v/  pqr^y/p  H"  V  9  +  %/ 0 - 
But,  since,  from  what  has  been  above  laid  down,  we  have 

V  P-i-\'iJ-\-  y/r  z=zx,  and  '^'pqr  =  \/h, 
d  these  be  put  for  their  equals  in  the  last  equation,  it  vpill  become,  by  this 
jiubstilulion, 

X*-\-2fx2—8h2x-\-f2—ig=0. 

Whence,  comparing  these  coefficients  with  those  of  the  given  equation, 
there  will  arise 

2/L=a  ;  — 8\//i=6  ;  f2—gc=  c,  or, 
-      a     ,       62  a2     C 

And,  consequently,  by  substituting  these  values  in  the  assumed  cubic  equa- 
tion (2),  we  shall  have 

1  63 

2/3  4.  ^ay2  ^  _  (ai_4c)3/«=  —  (3), 

the  three  roots  of  which  last  equation,  when  substituted  for  p,  q,  and  r,  ip 
the  formula  x=  A/p-f-  x/r-|-^/9,  will  i^ive,  by  taking  each  term  of  the  ex- 
pression both  in  _|_  and  — ^all  the  four  values  of  x. 

Or,  in  order  to  render  this  result  more  commodious  in  practice,  by  freeingf 
it  from  fractions,  let  y  =siz.  Then  by  substitution  and  reduction,  W*  shall 
have  the  corresponding  equation 

^2 


174 


BIQUADRATIC  EQUATIONS. 


Find  the  three  roots   of  the    cubic  equation  2^'\-2az^-t' 
(a^'-"ic)z=b^,  by  one   ot  the   former   rules,   before   given 
for  this  purpose;  and  let  them  be  denoted  by  v-',  r ",  andr'S 
Then,  we  sliall  have 


When  b  i.s  positive, 
_  —  ^/r  —  y/r"  —  ^r" 


-^/r'-hyr"-}-^r" 


2 


When  b  is  negative, 

_+x/r'+y/r"-\-^r'" 

2        ' 

_  +  ^/r'  —  x/r"  —\/r" 
2 

2 
_  -  ^r'      ^r"-\'x/r" 


2  \      ^  2 

Note.  If  the  three  roots  r',  r",  r'",  of  the  auxih'ary  cu- 
bif  equation  be  all  real  and  positive,  the  four  roots  of  the 
proposed  equation  will  al«o  be  real  ;  and  if  one  of  these 
roots  be  positive,  and  the  other  two  imaginary,  or  both  ot 
them  negative,  and  equal  to  each  other,  two  of  the  roots 
of  the  given  equation  will  be  real,  and  two  imaginary  : 
which  are  the  only  rases  that  produce  real  results. 

3.  Given  x^  — 25a:'-+6ua;--3b=0,  to  find  the  four  root? 
of  the  equation. 

Here  a—  -  25,  b~QO,  and  c=  —  36  ; 
Whence,  by  substituting  these  values  for  their  equals,   iu 
the  cubic  equation  above  given,   we  shall   lave  2^-2x25 
e2  -f  (252  _|_  4  x36)2  =  60",  or  2"^-.  50^^  +  l^^z  =  3600  : 


5r3  -}.2az2  -j-(a2— 4r)z=62,  (4) 
t!.e  three  roots  of  which  are  each,  evidenily,  four  times  those  of  the  former. 
Hence  using  (his  instead  of  equation  (3),  and  denoting  its  roots  bj  y',  r'',  v  , 
the  last  mentioned  formula,  taking  each  of  its  terms  in  -\~  and  — ,  as  before, 
\vill  give  the  values  of  x,  as  in  the  above  expressions. 

JVote.  If  we  were  to  take  all  the  possible  changes  of  the  signs,  in  this  case, 
which  the  terms  of  the  assumed  formula  admit  of,  it  would  appear  that  x. 
should  have  eight  different  values  ;  but  it  is  to  be  observed,  that  according 
to  the  first  part  of  the  above  investigation,  the  product  v/pX  v/?  X  \/?*  = 
V  h,  or  "5^6  ;  and,  consequently,  that  when  b  is  positive,  either  all  the  three 
radicals  must  be  taken  in  -|-,  or  two  in  —  and  oAein  -|- ;  and  when  b  is  nega- 
tive, they  must  either  be  all  — ,  or  two  4-  and  one  •— ;  which  consicleratioo? 
reduce  the  number  of  roots  to  four. 


BIQUADRATIC  EQUATIONS.  175 

the  three  roots  of  which  last  equation,  as  found  by  trial, 
or  by  one  of  the  former  rules,  are  9,  16,  and  25,  re-. 
spectively  ;  whence 

x=U- ^9—^16    V25)=i(— 3— 4— 5)=~6 

a:=,^(-4-v/94-v^l6-^5)-^i(4--^  1-4-5) -H-l 
And  consequently  the  tour  roots  of  the  proposed  equa- 
tion  are  1,  2,  3,  and  — 6. 

EXAMPLKS  FOR  PRACTICE. 

1.  Given  a;'^ -55 r"  —  30x-f- 504=0,  to  find  the  four 
roots,  or  values  of  x.  ilns.  3,  7,  — 4,  and  —  6» 

2.  Given  x'-'vIx'—lx^—Sx^^l:!,  to  find  the  four 
roots,  or  values  of  x.  Ans.  1,  2, — 3,  and  —2, 

3.  Given   x^-8.c'+  I4x=^+4x=8,  to  find  the  four  roots, 

or  values  of  x.  .  Ans.  \  '^X^^^'  ?  "^t 

}  I4-v/3,  1-a/^ 

4.  Given  x^ — i7x^-^20x-—6=O,   to  find  the  four  roots, 

or  values  of  x.  Ans.  \       ^f  ^I'     o""^^* 

5.  Given  x^~-'dx^~-4x^==''>,  to  find  the  four  roots,  or 
values  of  X.  Ans.    \      ^t^^   '?'     2~lv/   '^ 

6.  Givenx^~19.T^-l-l3'2J.i:^-3u^a?-f-2uO=0,  to  find  the 
•  'oots,  or  values  of  x.    Ans.  )  +^:f,6r]«^^^(_-S6^ 

7.  Given  a:^~27a;'^-'r  16  2a;H  3  36r- 1200— 0,  to  find 
U>efou„o„ts,orva..e.or..A„s.),^:°-08,-3.0^^^^^^ 

8.  Given  x'* — I2j;'^-f-  12j — 3=0,  to  find  the  four  roots,  or 

values  of  :r  Ans   ^    •'^"^OIS,   -3.907378 

^''^''^^  ^^  ^-  ^""-  ^  2.858083,  .443277 

9.  Given  .x*--36i;24-72x-^.3d  =  0  to  find  the  four  roots, 
or  values  of  T  Ans  J  <^-872'^S86,  1.2879494 
or  values  ot  r.  Ans.  j  ^  -3O0506,-6.S729836 

10.  Given  a:^~.12x3-f47x=^-72x+3tj  =  0,  to  find  the 
roQts,  or  values  of  x.  Ans.  1,  2,  3,  and  6, 


176  RESOLUTION  op  EQUATIONS 

11.  Given   x^-{-24x='— lUx^— 24x-f  «=0,    to   find  the 
roots,  or  values  of  x,  AnK.  ^  ^'^\l]Z  \l\  It'^yl 

12.  Given  .r'~  Cx^-SSx^—l  14.x- 11=0,   to    find  the 
roots,  or  values  of  x. 

Ans.  ±tv/3+f:tv>(17±VV3> 

OF  THE 

RESOLUTION  OF  EQUATIONS 

BY  APPROXIMATION. 


Equations  of  the  fifth  power,  and  those  of  higher 
dimensions,  cannot  be  resolved  by  any  rule  or  algebraic 
formula  that  has  yet  been  di>covered  ;  except  in  som« 
particular  cases  where  certain  relations  subsist  between 
the  coefficients  of  their  several  terms,  or  when  the  roots 
are  rational ;  and,  for  that  reason,  can  be  easily  found  by 
means  of  a  few  trials. 

In  these  cases,  therefore,  recourse  must  be  had  to  some 
of  the  usual  methods  of  approximation  ;  among  which 
that  commonly  employed  is  the  following,  which  is  univer- 
sally applicable  to  all  kinds  of  numeral  equations,  what- 
ever may  be  the  number  of  their  dimensions,  and  though 
not  strictly  accurate,  will  give  the  value  of  the  root  sought 
to  any  reqiiired  degree  of  exactness. 


Find,  by  trials,  a  number  nearly  equal  to  the  root 
sought,  which  call  r  ;  and  let  z  be  made  to  denote  the  dif- 
ference between  this  assumed  root,  and  the  true  root  x. 

Then  instead  of  a:,  in  the  fjiven  equation,  substitute  its 
equal  r±:z,  and  there  will  arise  a  new  equation,  involving 
only  z  and  known  quantities. 


BYAI*t»ROXlM4TlON.  HT 

Reject  all  the  terms  of  this  eq'iation  in  which  z  is  of 
Uvo  or  more  cliin«"nsi<»ns  ;  and  r  le  approximate  value  of 
z  may  then  be  df^t.^rmined  by  mt^ns  of  a  simple  equation. 

And  if  the  valuH.  thus  foimd,  I)  •  added  to,  or  subtracted 
trom  that  of  r,  acordin,^  as  r  wa^  a-:>««rned  too  little  or  too 
great,  it  will  :iive  a  n^ar  v  tlue  of  fiie  root  required 

But  as  this  approximation  will  seldom  be  sufficiently 
exact,  the  oporatioti  iniis=  b^j  repMf  ■  1  !»v  substitutuiji  the 
number  thus  foind  for  r  in  thn  iiMliTHd  equation  exhi- 
bilinij  the  value  of  z  ;  when  a  se;- >ud  ^-orrection  ot  z  will 
be  obtained,  vhicb,  bemg  added  ro,  or  subtracted  from  r, 
will  ^ive  a  nearer  value  of  the  rooi  fuaii  the  former. 

And  by  asjain  substitutinj;  this  ia.si  number  for  ,  in  the 
above  montio  led  equation,  asid  rep  atmii  t.ie  same  process 
as  often  as  may  be  thouifht  necessarv  a  value  of  re  maybe 
found  to  any  deirree  of  accurary  rfqiin^d. 

Note.  The  deci.nal  part  of  tn-  root,  as  found  both  by 
this  and  the  next  rule,  will,  in  ijMif^ral,  about  double  itself 
at  each  op'^ration ;  and  therefor.:?  if  wtiild  be  useless  as 
well  as  troublesome,  to  use  a  fn  ich  j/reater  "number  of 
figures  than  these,  in  the  several  substitutions  for  the 
values  of  r.* 

EXAMPLES. 

•^-I.  Given  a?^+x2  4-2;=90,  to  find   the   value  of  x  by  ap~ 
proximation. 

Here  the  root,  as  found  by  a  {q^^  trials,  is  nearly  equal 
to  4. 


*  it  mriy  h« m^  be  observed,  (hat  if  any  of  the  roots  of  an  equation  be 
whole  nmnbp  •>  tiiey  rnav  he  df>tprinined  bv  substituting  1,  2,  3,  4,  &-c.  suc- 
cessively, bofti  ill  -plus  and  in  minus,  for  the  unknown  quantity,  till  a  result  is 
obtained  equHi  to  that  in  the  question  ;  when  those  that  are  found  to  succeed, 
will  be  t;ie  roots  required. 

Or,  since  the  last  term  of  any  equation  is  always  equal  to  the  continued 
productof  all  its  roots,  the  number  of  these  trials  may  be  generally  diminish- 
ed, by  tinditi?  all  the  divisors  of  that  term,  and  then  substituting  them  both  in 
plus  and  minus,  as  before,  for  the  unknown  quantity,  when  those  that  give 
the  proper  result  will  be  the  rational  roots  sought ;  but  if  none  of  them  are 
found  to  succeed,  it  may  be  concluded  that  the  equation  cannot  be  resolved 
hy  this  method  ;  the  roots,  in  that  case,  being  cither  irrational  or  imaginary. 


1^  RESOLUTION  op  EQUATIONS 

Let  therefore  4  =  r,  and  r-h^=:jj 

Then     x''=r'^-^2r2+z^  =90. 

X  =r  4-2 
And  by  rejecting  the  terms  z^,  Srz'  and   z",  as  small  m 
comparison  with  z^  we  shall  have 

r^+H+r  +  3r2?+2r2+^,  =90  ; 
^,  90-r^— r^-r      90-64       1^-4       6 

Whence  ^^«=— — -  -  _  -  —  =  — — — 1 = — =.  10. 

Sr^-h   r+l  4h+84l  67 

And  consequently  x  — 4. »,  nearly. 
Again,  if  4.1  be  subsntuied  in  the  place  of  r,  in  the  last 
equation,  we  shall  have 
_90-r  -r2—r_90— 68.921  — 16.81— 4.1 

And  consequeniiy  x=4.  i  -{-.Oi  283=4. 10283  for  a  se- 
con'^i  .ipproxiiitatiofi 

And  if  the  first  four  figures,  4.102,  of  this  number  be 
again  substituted  for  r.  in  the  same  equation,  a  still  near- 
er value  of  the  root  will  be  obtained  ;  and  so  on,  as  far  as 
may  be  thought  necessary. 

2.  Given  a^4-^0a;=l0l»,  to  find  the  value  of  r  by  ap- 
proximation. Ans.   x=4. 142  1356. 

3.  Given  x^-}-9j^-\-Ax~Q0,  to  find  the  value  of  x  by 
approximation.  Ans.  x=2. 4721359. 

4.  Given  j' — 3«r'4-2I0x2-{-538a-f  i'89=0,  to  find  the 
value  of  X  by  approximation. 

Ans.  a:=.-!0.5366n375. 

5.  Given  x^-|-6.t^— 10"— 1 12x'--207.r-fll0=0,  to  find 
the  value  of  x  by  approximation. 

Ans.  4.46410161. 
The  roots  of  equations,  of  all  orders,  can  also  be  de- 
termined, to  any  degree  of  exartnesn,  by  means  of  the  fol- 
lowing easy  rule  of  doiihlp  position  ;  which,  though  it  has 
not  been  generally  employed  for  this  purpose,  will  be 
found  in  some  respects  superior  to  the  former,  as  it  can 
be  applied,  at  on^e,  to  any  ut>reducecl  equation  consisting 
of  surds  or  compound  quantities,  as  readily  as  if  it  had 
been  brought  to  its  usual  form. 


BY  APPROXIMATION.  179 


RUL£  II. 

Find,  by  trial,  two  numbers  as  near  the  true  root  as 
possible,  and  substitute  thetn  in  the  i^ivon  equation  instead 
of  the  unknuwn  quantity,  noting  the  results  that  are  ob- 
tained from  each. 

Then,  as  the  difference  of  these  results  is  to  the  differ- 
ence of  the  two  assumed  numbtrs,  so  is  the  difference  be- 
tween the  true  restilt.  ^iven  by  the  qu»^stion,  and  either  of 
the  former,  to  the  correction  '>f  the  nu  iber  belonging  to 
the  result  used  ;  which  correction  bein^  added  to  that 
number  when  it  is  t«)o  little,  or  subtracted  from  it  when  it 
is  too  great,  will  give  the  root  required  n>arly. 

And  if  the  number  thus  determined,  and  the  nearest  of 
the  two  former,  or  any  other  that  appears  to  be  more  ac* 
curate,  be  now  taken  as  the  assumed  roots,  and  the  opera- 
tion be  repeated  as  before,  a  new  value  of  the  unknown 
quantity  will  be  obtamed  still  tnoro  correct  than  the  first  ; 
and  so  on,  proceeding  in  this  manner  as  far  as  may  be 
judged  necessary.* 

*  The  above  rule  for  Uoublf  P  is.iiou,  which  is  much  more  simple  and 
commodious  than  the  one  commonl-,  employed  for  this  purpose,  is  the  same 
■as  that  which  was  first  given  at  p.  Jll  ot  the  octavo  edition  of  rny  Arithme- 
tic, published  in  1810. 

To  this  we  may  farther  add,  that  when  one  of  the  roots  of  an  equation  has 
been  found,  either  hy  this  method  or  the  former,  the  rest  may  be  determined 
as  follows  : 

Bring  all  the  terms  to  the  Ir-ft  hand  side  of  the  equation,  and  divide  the 
whole  expression,  so  formed,  bv  the  diffcience  between  the  unknown  quanti- 
ty {x)  and  the  root  first  found  ;  and  the  resulting  equation  will  then  be  de- 
pressed a  degree  lower  than  the  i^iveii  one. 

Find  a  root  of  thi=  eijUciiiofi.  by  approximation,  as  in  the  first  instance, 
and  the  number  so  obtained  will  he.  a  second  root  of  the  original  equation. 

Then  by  means  of  this  root,  and  the  unknown  quantity,  depress  the  se- 
cond equation  a  degree  lower,  and  thence  find  a  ihird  root  ;  and  so  on,  till 
the  equation  is  reduced  to  a  quadratic  ;  when  the  two  roots  of  this,  together 
with  the  former,  will  be  the  roots  of  the  equation  required 

Thus  in  the  equation  .t3 — \r,x2  -j_63a;  =  50,  the  first  root  is  found  by  ajx- 
pi'oximation  to  be  1.028U4      Mence, 
v— 1.02804)2; 3— 153:2  -}-G3r— 50(x2— ]3.97195x'-{-48  63627=0. 

And  the  two  roots  of  the  quadratic  equation,  x2. — 13.9719Sa?  =  — 48.63627 
found  in  the  usual  way,  are  G    76i.>  and  7.39543. 

So  that  the  three  roots  of  the  -iven  cubic  equation  a; 3 — 15a:a  -f- 63x  =  50, 
are  1.02304,  6.57C53,  and  7  39543,  their  sum  being  c=  15,  the  coefficient  of 
the  second  term  of  the  equation,  as  it  ought  to  be  when  they- are  right. 


RESOLUTION  of  EQUATIONS 


EXAMPLES. 

K  Given  a::^+a;^+x=100,  to  find  an  approximate  value 
©f  X. 

Here  it  is   soon  found  by  a  few  trials,  that  the  value  of 
X  lies  between  4  and  5. 

Hence,  by  taking  these  as  the  two  assumed  numbers,  the 
operation  will  stand  as  follows  : 

First  Sup.  Second  Sup. 

4     .     .     X      .     .  6 


16     . 

.     x^     . 

.        '25 

64     . 

.      a-3     . 

.      125 

84 

Results 

165 

155     . 

.     5      . 

.      109 

Therefore 

84     . 

.      4       . 

.        84 

71 

:      1      :: 

16 

.225. 
And  consequently  a-=44-. 225=4. 225,  nearly. 
Again,  if  4.2  and  4.3  be  taken  as  the  two  assumed  num- 
bers, the  operation  will  stand  thus  : 

First  Sup.  Second  Sup. 

4.2  .      .     X      .     .  4.3 


17.64 

74.088 


Therefore 


95.928 
102.297 
P5.928 


Results 
.  4.3 
.     4.2 


18.49 
79.507 

102.297 
102.297 
100 


6.369     :         .1      :  :     2.297     :      .036. 
And  consequently  x=4.3 — .036=4.264,  nearly. 
Again,  let  4.264  and  4.265  be  the  two  assumed  num 
bers  ;  then 


First  Sup. 

4.264 
16.181696 
77.526752 


Second  Sup. 
4.265 
ly.  190223 
77.581310 


99.972448     Results     100.036535 


^Y  APPROXIMATION. 


!8! 


Therefore 
00.036535     4.265      100 
99.972448     4.^«4     99.972448 


.0640«7  :     .Oui   ::     .U2.7552  :  .0004299 
And  consequently 
a;=4.264+.0004^99=4.  .^6442.99,  very  nearly, 
2.  Given    {^x^—  I5y-jrx^x  =  90j  to   find  an  approxi- 
mate  value  of  x. 

Here,  by  a  few  trials,  it  will  beigoon  found,  that  the  va- 
lue of  X  lies  between  10  and  11;  which  let,  therefore^ 
i)e  the  two  assumed  numbers,  agreeably  to  the  directions 
given  in  the  rule. 

Then 
First  Sup.  Second  Sup, 


Hence 


25      .     .    ( 
31.622  .  . 

56.622 
121.122  .  . 
56.622  .  . 


lx--I5)^ 
x^% 

Resul'is 
11.     . 
10  .     . 


84.64 

.  3b.4«2 

121.122 
121.122 
90 


64.5         :        1     ::         3 1.122:. 482. 

And  consequently  a;=ll — .482  =  10.518. 

Again,  let  10.5  and  10.6  be  the  two  assumed  numbers. 

Then 

First  Sup.  Second  Sup. 

49.7025  .   .   (ia;2--15)2  .   .  55.8.'^0784 

34.0239  .   .  x^x    .  .  34.511099 


83.7264 

.  Results  .  . 
Hence 

90.341883 

90.341883  .  , 

10.6     . 

.  90.341883. 

33.7264       .  . 

10.5     . 

.  90. 

6.615483     :  .1      ::      .341883:0051679 

And  consequently 
r=i0,6  -.0051679=10.5948321,  very  nearly. 


182  EXPONENTIAL  EQUATIONS. 

EXAMPLES  FOR  PRACTICE. 

1.  Given  x^+lOx'-}- 5a: =2600,  to  find  a  near  approxi- 
mate  value  of  .r.  Ans.  x=  11.00673. 

2.  Given  2x*-- 16x^+40x2— 50x4-1=0,  to  find  a  near 
value  of  X.  Ans.  cX=  1.284724. 

3.  Given  a;'4-2x^+3x^+4.T'+5a.=54b2l,  to  find  the 
value  of  x.  Ans.  8.414465. 

4.  Given  y(7x^+4x2)  +  ^(20x2— I0a)=2t5,  to  find  the 
value  of  X.  Ans.  4.510661. 

5.  Given  v/(«44x2— (x24-20)2)+v/(i96x2— (^'+24)2) 
=  11 4,  to  find  the  value  of  x.  Ans.  7. 1 23S83. 

Of  exponential  EQUATIONS. 

An  exponential  quantity  is  that  which  is  to  he  raised  to 
some  unknown  power,  or  which  has  a  variable  quantity  for 
its  index ;  as 

1  i 

a^,  a^,  x^,  or  x''",  &c. 

And  an  exponential  equation  is  that  which  is  formed  be- 
tween any  expression  of  this  kind  and  some  other  quanti- 
ty, whose  value  is  known  ;  as 

a^=6,  x^=a,  &c. 

Where  it  is  to  be  observed,  that  the  first  of  these  equa- 
tions, when  converted  into  logarithms,  is  the  same  as 

T  log.  a=6,  or  x=  .  ;  and  the  second    equation  x^= 

log.a  ^ 

a  is  the  same  as  x  log.  x=log.  a. 

In  the  latter  of  which  cases,  the  value  of  the  unknown 

quantity  x  may  be  determined,  to  any  degree  of  exactnes.'^ 

by  the  method  of  double  position,  as  follows  : 

RULE. 

Find,  by  trial,  as  in  the  rule  before  laid  down,  two  num- 
bers as  near  the  number  sought  as  possible,  and  substitMt> 
them  in  the  given  equation 

X  loff.  x=lo£.  a. 


EXPONENTIAL  EQUATIONS.  185 

matead  of  the  unkaown  quantity,  noting  the  results  obtain 
cd  from  each. 

Then,  as  the  difference  of  these  resuhs  is  to  the  differ- 
ence of  the  tvvo  assumed  numb  rs,  so  is  the  difference 
between  the  tru»i  resuh,  ^iven  in  the  question,  and  eittier 
of  the  firmer,  to  the  correction  o>  the  nmnber  belonging 
to  the  resuU  used  ;  which  correct!  n  bem^  added  to  that 
nuinber,  when  »r  is  too  httln,  or  Mr.>'racted  from  it,  when 
it  is  too  great,  will  ^'\ve  the  -oot  requi'nd,  nearly. 

And-  if  tiie  na(ni)er  thus  detei  muuhI.  and  the  nearest 
of  the  tvvo  former,  or  any  other  tiin'  apfiears  to  be  nearer, 
be  taken  as  the  assumed  roots  and  ine  operation  be  re- 
peated as  bef  .re,  a  new  vahie  o}'  tne  unknown  quantity 
will  be  obtained  still  more  correct  man  the  first;  and  so 
on,  proceeding  in  this  manner,  a>  far  as  may  be  thought 
necessary. 

EXAMPLfS. 

1.  Given  a;''=100  to  find  an  approximate  value  of  x. 
Here,  by  the  above  formula,  we  tiave 

X  log.  X  =  lotr.    li)U=2. 

And  since  x  is  readily  found,  by  a  few  trials,  to  be  nearly 
in  the  middle  between  S  and  4,  but  rather  nearer  the  lat- 
ter than  the  former,  let  -».5  and  3.6  be  taken  for  the  two 
assumed  numbers. 

Then  lojr.  3.=-  44'>*;80,  which,  being  multiplied  by 
3.5,  gives  1.9042   S  =  rtrst  result; 

And  log,  3.6=. 5  .630^:0,  which,  being  multiplied  by  3.6, 
gives  2.0U26s»y  for  the  second  result. 
Whence 
2.00:^689  .  .  3.6  .  .     .002639 
1.904-238  .  .  3.6  .  .  t. 


.t»9845!    :       .1     ::    .002689  :  .00273 
for  the  first  correction  ;  which,   taken   from   3.6,  leaves 
Tn=3.o97ii7.  nearly. 

And  as  this  value  is  found,  by  trial  to  be  rather  too 
small,  let  3.59727  and  3.53728  be  taken  as  the  two  assum- 
ed numbers. 


184  BINOMIAL  THEOREM. 

Then  log.  3.59728=0.555974243184677  to  15  piacef 
The  log.  3.59727=0.555973()351'47267  to  15  places 

which  logarithms,  multiplied  by  their  respective  numbers 

give  the  following  products  : 

1.999985  226S229&  I  ^^^^  ^''"^  ^^  ^^^  ^^"^  ^S"'^" 
Therefore  the  errors  are  4974656488 

14877337702 
and  the  difference  of  errors  9902681214 

Now  since  only  6  additional  figures  are  to  be  obtained, 
we  may  omit  the  three  last  fiirures  in  these  errors ;  and 
State  thus:  as  difference  of  errors  9902681:  difference  of 
sup.  J  : :  error  4974656  :  the  correcfion  502354,  which 
united  to  3.59728  gives  us  the  true  value  of  ar— 
3.59728502354.* 

2.  Given  3:^=2000,  to  find  an  approximate  value  of  .r. 

Ams.  a: =4.82782263. 

3.  Given  (6t)'^=96,  to  find  the  approximate  value  of  a- 

Ans.  0,=  1.8826432. 

4.  Given  a;  =123456789,  to  find  the  value  of  j:. 

Ans.  8.6400268. 

5.  Given  a:''  —  x  =  (2x — x'^y,  to  find  the  value  of  or. 

Ans.  a:=  1.747933. 


BINOMIAL  THEOREM. 

The  binomial  theorem  is  a   general  algebraical  expres- 
sion or    formula,  by  which   any  power,    or  root  of  a  given 
quantity,  consisting  of  two  terms,  is  expanded  into  a  series  : 
the  form  of  which,  as  it   was  first  proposed   by   Newton 
being  as  follows : 


*  The  correct  answer  to  this  question  has  been  first  gireu  by  Doctor 
Adrain,  in  hi?  edition  of  Huftoii's  Maihematif-,  w')o  plainly  proves  that  Hut 
ton's  answer,  which  is  th--  same  as  Boimv  castle'?,  i?  incorrect  :  See  Hvtfon^^ 
Mathematics,  Vol.  I.  p.  263.  JS".  Y.  Kdition-  Ec 


BINOMIAL  THEOREM.  185 

m       m         ^       rn  /m — n\        .  m  /m — n\ 
'  ^       n         n  \  2n    f  n  \  2n    f 

Or, 

»/!       »«     m  ■m  —  n       ,  m  -  2/i       , 

,P  +  PQ)n  =Pn  +— AdH BQi CQ  + 

m  —  3/1 

DQ,  &c. 

Where  p  is   the   first  term  of  the  binomial,   Q,  the  second 

III 
term  divided  by  the  first,  —    the  index   of  the   power,    or 

^  n 

root,  and  a,  b,  c,  &c.  the  terms  immediately  preceding 
those  in  which  they  are  first  found,  including  their  signs 
'f-or-. 

Which  theorem  m?iy  be  readily  applied  to  any  particu- 
lar case,  by  substitutmg  the  numbers,  or  letters,  in  the 
^iven  example,  for  p,  q,  m,  and  n,  in  either  of  the  above 
formulse,  and  then  findmg  the  result  according  to  the  rule.* 

*  This  celebrated  theorem,  which  is  of  iht-  most  exteiiaive  use  iti  al^^ebra, 
and  various  other  branches  of  analysis,  may  be  otherwise  expressed  as  fol- 
lows : 

Or,  (a-|-a)p= 
n^  "^n  a-j-x      n    2n      a-f-x  n      'Zn        3rt      o^a 

771 

Or,  (a+a?)-  = 

.,  "^r.     "^,« — x^  ,mm-^na — x       mm-i-n  m-l-2na—x^   .  ^ 

i2a-[l ( )H — .    - — ( )2 .— - — .  —  _(— -_)3]  &c.  . 

n^       na-i-x      n    2n     a -^x       n      ^n.        3n     a+x 
It  may  here  also  be  observed  that  if  ?n  be  made  to  represent  any  whole,  oi 

fractional  number,  whether  positive  or  negative,  the  first  of  these  expression? 

may  be  exhibited  in  a  more  simple  form 

,     ,    vw         m  ,        m-i     ,  m(m— I)  „,_2    o    .  m(/n— 1  )(/n— 2)  ™.3    3 

(a+x)    =a    -\-ma      x-] — i— 3"-«       *    -1 r ;; — ;r~  "     ^  * 

m{m — l)(m— 2) [m — (n — \)]a    ~  x 


1.2.3.4      .      ...  n 

Where  the  last  term  is  called  the  general  term  of  the  series,  because  if  1 ,  2. 
■3,  4,  (fcc.  be  substituted  successively  for  n,  it  will  give  all  the  rest, 

r2 


186  BINOMIAL  THEOREM, 


EXAMPLES. 

1.  It  is  required  to  convert  {a^-\'.r)^  into  an  infinite  se 
ries. 

TT  o  3"        *'l  1 

Here  2—a%  Q=-— ,  -=^,  or  m  =  I,  and  w.  =  2  •- 
whence 

m  m  I 

Pn"=(a-}n  =(a2)2=:a=^^ 

wi  1      a      a       a- 

-ACl  =  -X-X— =^r-=B, 

m — n      __1~2      X       x_       x^ 

m—2n     I — 4  x^        x         3x^ 

^w~3n      __  1  -  6        3r3  a- _  5.5.t^      __ 

4n     "^        8         i?.4..6^^^^  "  2.4.6.Sa'~^' 

6n       ^~~    10        ""2^76^ 8a t"     o^""  2.4.6. S.lOa"""* 
&C.  &c.  J         &c. 

Ttierefore  (n^'{-x)^== 

2a     2.4a^'^2.4.6^""  2A.6.8a''^2A~6.8A0a''^'  ^^ 
Where  the  law  of  formation  of  the  several  terms  of  i\. 
series  is  sufficiently  evident. 

2.  It  is  required  to  convert  T-rrro>   or  its   equal   (a 4 
by^j  into  an  infinite  series. 

Here  p=a,  ci=-,  and  -= — 2,  or  w=-2,  and  n=l 
a  71. 

whence 

p„=i(a)n=a      = — =:a, 

m             2      1      b         2b 
~Aa=-7X— -X-= r=B. 


BINOMIAL  THEOREM.  18^ 

tn^n         —'2—1         2h     b     W 

'-  — BQ  =  — - — X -X-=— =c, 

2w  2  a^     a      a^ 

m^2n  -2-2     3^2      6         Ab^ 

-  ^    -CQ=- X-^X-= r-^D, 

m—^n          —2-3  46  ^      f,     3b* 

4n                    4  a^       a       a^ 

&c.                       &c.  &c. 

,1            I       2,>>  .  362      463     5^4 
Consequently  - — ---——- --\ — -  --_  +  _     &c. 

3.  It  is  required  to  convert —,    or    its     equal    a'^ 

{a?—x)^^  into  an  infinite  series. 
Here 

P=a^  Q= .and -=-?:-,    or   y«— —  1,   and    m=L»  ; 

whence 

'a 

-^Aq         -X-     ""^  —  ^3""^' 

m  —  n          — 1—2       X             oc        3x^ 
.Bq= X— -X -=■ 


2n  4  2a  3  ^2      2.4a' 

m-2n  —1—4      3a;2  rr         3.5i;^ 

CQ=  _X— -3  X -—=—-— =D, 


3w  6  2.4a5         rt2      2.4.6a 

'  4n~°^  8~     2X6^^      a2""2.4.6.8a^~^' 

&c.  &c.  &c. 

Therefore 
-_i_i=l_L.V^>14.1  .^-S.    3»5/^%  3.5.7     x^, 
(a^-a-)^     o"''2^a3>''^2.4Vi5>'^2.4.6'<a'^^2.4.6.8^a9/ 
&c. 

And 

&c. 


188  BINOMIAL  THEOREM. 

4.  It  is  required  to  convert  ^9,  or  its  equal   (S-f-J) 
into  an  infinite  series. 

■WW  I  ,   771         I  -,  1  o 

Here  p=S,  (i=-  and  -  =  -,  or  m=l  and  n=3  ; 

b  71       3 

^  !?       ,  Whence 

P«=(S)"=S^  =  2  =  A, 
m  I      ^2       1  I 

,n_7i         1—3       :        1  1 

in  —  2n  1—6    ,  1  >  5 

3?i  !<  3.6.2'      c>*      d.6.9.1'' 

?n  — 3;i  ]-9  5  1  5.S 


4a  12       3.6.y.2'      ii^  3.6.9.12.2^° 

m-4n      _1  — 12  5.8         ^     i   __  5.S.  11 

___EQ         ^-      — 3^579^  ^g.^Ji^'^  23  ~3. 6. 9.12.1572' 

&c.  &c.  &c. 

Therefore  i/9  = 

^"^372^""  3^6r2^"^3.6  972' "3.6.9.12.2"^ '^3X9T«7l"5^2' 

&c. 

5.  It  is  required  to  convert  ■y/2,  or  its  equal  ^{\-\-\]. 
into  an  infinite  series. 

1        13  3  5  3.5.7 

^"'-  ''^2-2l"^274-76-^7:rH78+2X678To^^;- 

6.  It   is  required  to  convert   ^7,  or  its  equal  (8  -  1)". 
nto  an  infinite  scries. 

«        116  5.8  . 

3.^2-     3.6.^2^     3.6.9.2^     .5h.9.l2.2^« 

7.  It   is    required   to    convert     ^240,    or     its    equa 

(243 —  3)^,  into  an  infinite  series. 

1  4  4.9  4.9.14 

\r^c     ^ ..— — &C 

5.3^   5.10.3^     5.  It).  15.3"     5. 10.15.20.3^^ 

8.  It  is  required  to  convert  {a+x)-  into  an  infinite  series. 
4ns.  a-  I  ,±_.^^^j.__«^_-_4:&c.  ^ 


BINOMIAL  THEOREM.  189 

9.  It  is  required  to  convert  (a  ±6)3  into   an  infinite  se- 
ries. 

Ans.  a3  }  i-t- r _-f-&c.  l 

(         Sa     S.6a'     3.6.9.a3       3.6.9. I2a^  "  S 

i 

10.  It  is  required  to  convert  (a —  6)*    into    an     infinite 

series 

'     iC  b        Sb^  S.-b^  3.7.116^  „       > 

Ans.  a*  ^  1 &c.  J 

^         4a     4.Ha-     4.-,.\2<i^     4.8..2.  l6a*  > 

2_ 

11.  It  is   required  to  convert  (a+^c)^    into   an   infinite 
series. 

Ans.o3  7  i-i- 4- — -— &c.  } 

12.  It  is  required  to   convert  (1—a;)^    into   an    infinite 
series. 

2x     2.3a;2     2.S.8x'      2.S.8.133;'* 
Ans.  1  _____ -.  _^_  - ^j(j  j^20  " ^'^* 

13.  It   is  required   to   convert -^   or    its    equal 

—1 

ra±x)    2  into  an  infinite  series 

I  c  T  ;S.r2         3.5.r'  3.5.7x*'      _         > 

^"^-  T  '  ^2;i^2.-^  ^2.4:6^-^^iX6:s:;?^^"*  ^ 

a^  ^ 

14.  It  is   required    to   convert j-,    or     its   equa? 

(a±x)3 

art.T)~2  into  an  infinite  seaes. 
.  f  ^  ,_x  4r2         4.7x''     ,     4.7.10a;^       _        ) 

J  5.  It  is  required  to  convert j,  or    its    equat 

— JL 

'  I  -{-x)    ^  into  an  infinite  series. 

X  ,     6t-        e.lla;^    .  6.11.16c*        . 

-^"^'  ^"5+5;R>--5TTo7i5+5jo:r5:2o-^'^ 


190  INDETERMINATE  ANALYSIS. 

I  %  or  its  equal  (a 

-|-a?)^(a^— a;^)~^,  into  an  infinite  series. 

OF   THE 

INDETERMINATE  ANALYSIS. 

I.v  the  common  rules  of  Alyrebra,  such  questions  are 
usually  propijsed  as  rnquire  s«ime  certain  "r  definite  an- 
swer ;  in  which  ca<e,  it  is  necessary  that  there  should  be 
as  many  indtjpendent  eqiati  ins,  expressing  their  condi- 
tions, as  ther«  are  nnkoowri  quantities  t(i  be  determined  ; 
or  otherwise  the  proolem  »vould  not  be  liuiited. 

But  m  other  branches  of  the  science,  questions  fre- 
quently arise  that  invcilve  a  greater  number  of  unknown 
quantities  than  there  are  equations  to  express  them  ;  in 
which  instances  they  are  called  indeterminate  or  unlimit- 
ed problems  ;  beingf  such  as  usually  admit  of  an  indefinite 
number  of  solutions;  althouoh,  when  the  question  is  pro- 
posed in  integers,  and  the  answers  are  required  only  in 
whole  positive  numbers,  they  are,  in  some  cases,  contin- 
ed  within  certain  limits,  and  in  others,  the  problem  may 
become  impossible. 

Problem  1, 

To  find  the  inteajral  values  of  the  unknown  quantities  a 
and  y  in  the  equation 

a.T— o</=i:c,  or  ar-^bij=c. 

Where  a  and  b  are  supposed  to  be  given  whole  num- 
bers, which  admit  of  no  common  divisor,  except  when  it 
is  also  a  divisor  of  c. 

RULE. 

1.  Let  wh  denote  a  whole,  or  integral  number  ;  and 
reduce  the  equation  to  the  form 


INDETERMINATE  ANALYSIS.  191 

budtc    ,  c  —  by     , 

{c=^— why  or  x= tuft. 

a  a 

2.  Throw  all  whole  numbers    out  of  that  of  these  two 

expressions,   to    which  the    question   belongs,  so  that   the 

numbers  d  and  e  in  the  remaimng  parts,  may  be  each   less 

than  a  ;  then 

di/:±:e         .  e  —dy 

— =wh,  or -='wh. 

a  o 

3.  Take  such  a  multiple  of  one  of  these  last  formulae, 
aorresponding  with  that  above  mentioned,  as  will  make 
the  coefficient  ot  y  nearly  e«[ual  to  a,  and  throw  the  whole 
numbers  out  of  it  as  before. 

Or  find  the  sum  or   ditference  of—,  and  the  expression 

a 

ctv 
above  used,  or  any  multiple  of  it  that  comes  near  — ,  and 

CL 

the  result,  in  either  of  these  cases,  will  still  be  =tiy/i,  a 
whole  number. 

4.  Proceed  in  the  same  manner  with  this  last  result ; 
and  so  on,  till  the  coefficient  of  y  becomes  =1,  and  the 
remainder  =  some  number  r  ;  then 

=wh.=p,  and  y=ap.^r, 

Where  p  may  be  o,  or  any  integral  number  whatever, 
that  makes  y  positive  ;  and,  as  the  value  of  y  is  now  known, 
that  of  X  may  be  found  from  the  given  equation,  when  the 
«[uestion  is  possible*. 

Note.  Any  indeterminate  equation  of  the  form 

in  which  a  and  h  are  prime  to  each  other,  is  always  possi- 
ble, and  will  admit  of  an  infinite  number  of  answers  in 
whole  numbers. 

But  if  the  proposed  equation  be  of  the  form 

ax-\-by=c, 

*  This  rule  is  founded  on  the  obvious  principle,  that  the  sum,  difference, 
Mr  product  of  any  two  whole  numbers,  is  a  whole  number  ;  and  that  if  a 
number  divides  the  whole  of  any  other  number  and  a  pail  of  it,  it  will  also 
^.livide  the  remaining  part. 


192  INDETERMINATE  ANALYSIS. 

the  number  of  answers  will  always  be  limited  ;  and,  iii 
some  cases,  the  question  is  impossible  ;  both  ol  which  cir- 
cumstances may  be  readily  discovered,  from  the  mode  of 
solution  above  given*. 

EXAMPLES. 

1.  Given    19—  l4y=\l,  to  tiiid  x  and  y  in  whole  num- 
bers. 

14v+ll        ,  .    ,      19y        L 

Here  x= — —-  — =^ufi..  and  also =wn. 

19  ly 

l^W        4J/+11      5i/— 11  , 

Whence,  by  subtraction,  _-^--^-£— —  =-^— =a;/i. 

Li:'  t  y  id 

Also,  -^— -  -  X  4  =  — ^-      -  =  i,"  -  2  -f  ^— -  =wh. 

And  by  rejecting  «/  — 2,  which  is  a  whole  number, 
y —  b 
19                 ^ 
Whence  we  have  y=^\9p-\-6. 
I43/+II       14(19/^+    )  +  ll      2b6/>4-95 
And  0:  =  -^  _= =—79—^ 

14/^+5. 
Whence,  if  p  be  taken  =0,  we  shall  have   ^=5  and  y 
=6,  for  their   least  values  ;  the  nun»ber   of  solutions  be- 
ing obviously  indefinite. 

2.  Given  2x-\-'6y=2b,  to  determine  x  and  y  in  whole 
positive  numbers. 

*  That  the  ccefficients  a  and  6,  when  these    two   formulse   are  possib!. 
ohould  have  no  common  divisor,  whirh  is  not  at  the  ?ame  tJme,  a  divisor  of  r. 
is  evident;  forifa=ajmd,  and6'=  me,v,'c  shall  have  ax  ^h  63/  =:mdx^meys=^ 

c ;  and  consequently  dx-{-ey=^—.  But  d.  e,  x,  y,  being  supposed  to  be  whole 

c 

uurnbers,  —  must  also  be  a  whole  number,  which  it  cannot  be,  except  when 

m  *^-' 

'U  is  a  divi«or  of  c 

Hence,  if  it  were  required  to  pay  lOOZ.  in  guineas  and  moidores  only,  the 
ijuestion  would  be  impossible  •,  since,  in  ihe  equation  21x+27y  =  2000,  which 
represents  the  conditions  of  the  problem,  the  coefficients,  21  and  27,  are  eack 
divisible  by  3,  whilst  the  absolute  term  2000  is  not  divisible  by  it.  See  my 
Treatise  on  Algebra,  for  the  method  of  resolving  questions  of  this  kind,  by 
means  of  Continued  Fractions. 


1- 


SN  DETERMINATE  ANALYSIS.         193 

Here  x=^—^^=^\^^y-\—^- 
Hence,  since  x  must  be  a  whole  number,  it  follows  that 

y 

-  must  also  be  a  whole  number. 

Let  therefore  —~-=.wh=^p  j 

Then  i—y=2p,  or  y=i — -2^, 
And  since 

,=12— 2/+^-=12-(l-2;>)+/, =12+3^-1, 

We  shall  have  r  =  1 1  -\-3p,  and  y=l  —2p  ; 
Where  p  may  be  any   whole   number  whatever,  that  will 
render  the  values  of  x  and  y  in  these  two  equations  posi- 
tive. 

But  it  is  evident,  from  the  value  of  y,  that  p  must  be 
either  0  or  negative ;  and  consequently,  from  that  of  x, 
that  it  must  be  0,  —1,  — 2,  or  — 3. 

Whence,  if  p=0,  p=^—),p= — 2,j9=— -3, 

^^"  ^2^=1>    2/=3,  2,-5,  2/=7; 

Which  are  all  the  answers  in  whole  positive  numbers 
that  the  question  admits  of. 

3.  Given  Sx^Sy—lQ  to  find  the  values  of  a;  and  y  in 

whole  numbers. 

Sy-\0     «       .  ,  %— 1        .  %— 1        . 

Here  x=-^ =22/  — 5+-^^ — =^h  ;  or  -^- — =r.^f> 


3  ^         '       3  '  3 

Also  .I--X2=^-^^-=y+^=wk, 


Or,  by  rejecting  y^  which  is  a  whole  number,  there  will 

•    y--2 
iiemam  ■        z=wh.=^p. 

Therefore  2/=3/)4-2, 

-     ,        8y—\6     8(3;7+2)-16     24p     . 
And  0.=-^-= AX-^ =_L=8;,. 

Where,  if  p  be  put  =  1 ,  we  shall  have  a;=8  and  j=5. 


194  INDETERMINATE  ANALYSIS. 

for  their  least  values  ;  the  number  of  answers  being,  as  in 
the  first  question,  indefinite. 

4.  Given  2ia-|-17t/=^00(J,  to  find  all  the  possible  va- 
lues of  X  and  y  in  whole  numbers. 

„             2000—172/     ..r  ,  S— l^y 
Herea;== ._ — ^=95H _-^=a,/,.  ; 

5 i7y 

Or,  omitting  the  95,  — ——=^7&;h. ; 

2lw     t — iTy     4v+5 
Consequently,  hy  addition,  — -H 57~   — "4i — =^h*  ; 

Also.  X__x5=-X--=l+_-^=»^., 

4-4-201/ 
Or,  by  rejecting  the  whole  number  I,  — 57— =«''''•  '■> 

And,  by  subtraction,  — — -^=?L^=a;ft.=p  ; 

Whence  2/=21;i4-4, 

And  .=!2^°=±'l(=!^°^^(liP±!)=92-17^. 
21  21  '  ' 

Where  if  p  be  put  =0,  we  shall  have  the  least  value  ot 
V=4,  and  the  corresponding,  or  greatest  value  of  a'=92. 

And  the  rest  of  the  answers  will  be  found  by  adding  2 1 
continually  to  the  least  value  of  t/,  and  subtracting  17  from 
the  greatest  value  of  x  ;  which  being  done  we  shall  obtain 
the  six  following  results  : 


j:=92 

3/=4 


75  I  58 
25     46 


41      24     7 
67     88      109 


These  being  all  the  solutions,  in  whole  numbers,  that 
the  question  admits  of. 

NoU  1.  When  there  are  three  or  more  unknown  quan- 
tities, and  only  one  equation  by  which  they  can  be  deter- 
mined, as 

n2'-l-6i/4-ce  =  (/, 
it  will  be  proper  first  to  find  the  limit  of  the   quantity  that 
has  the  greatest  coefficient,  and  then  to  ascertain  the  dif- 
ferent values  of  the  former,  from    I   up  to  that  extent,  a.« 
in  the  following  question. 


INDETERMINATE  ANALYSIS.  196 

3.  Given  3r4-5|/+72r=100,  to  find  all  the  different  va- 
lues of  X,  y,  and  z   in  whole  numbers*. 

Here  each  of  tlio  least  integer  values  of  x  and  t/  are  1, 
by  the  question ;   whence  it  follows,  that 

lUv)     5—3     100— 8_92_ 
^= =^^ y-~13^. 

Consequently  z  cannot  be  greater  than  13,  which  is  also 
the  limit  of  the  number  of  answers  ;  though  they  may  be 
considerably  less. 

hy  proceeding,  therefore,  as  in  the  former  rule,  we  shall 
have 

3  <5 

And,  by  rejecting  S3—y — 20, 

llf|r.i=»ft. ;  or  ^+i^P^=^-±^--=»A.  : 

Whence  - — k—^P' 
o 

And  y=2'sp-\-z — 1  ; 

And  consequently,  putting  p=0,  we  shall  have  the  least 

value  ofi/=^--  I  ;  where   z   .T.ay  be  any   number.  fVcrr.  ) 

up  to  13,  that  will  answer  the  conditions  of  the  question. 

When,  therefore,  2=2  we  have  y=-ly 

.    ^  100-19      _ 

And  2:= ^=27. 

tj 

Hence,  by  taking  z=2,  3,  4,  6,  &c.  the  corresponding 

values  of  .1  and  y,  together  with  those  of  2,  will  be  found 

to  be  as  below. 


2=    2 

:^ 

4 

5 

6 

7 

8 

y=  » 

2 

3 

4 

5 

6 

7 

x=2l 

23 

19 

15 

11 

7 

3 

*  If  any  indeterminate  equation,  of  the  kind  above  given,  has  one  ormorfe 
of  its  coefficients,  as  c,  negative,  the  equation  may  be  put  under  the  form 

ax-{-  by  :z=td^cz, 
ia  which  case  it  is  evident  tliat  an  indefinite  number  of  values  may  be  given 
to  the  second  side  of  tiie  equation  by  means  of  thr  indefinite  quantity  z  ;  and 
consequently,  also,  to  x  and  y  in  the  first.  And  if  the  coefficients  a,  6,  c,  in 
any  such  equation,  have  a  common  divisor,  while  d  has  not,  the  question,  as 
in  the  first  case,  becomes  impossible. 


106  INDETERMINATE  ANALYSIS, 

Which  are  all  the  integral  values  of  a*,  y,  and  z^  tha^ 
can  be  obtained  from  the  given  equation. 

JVote2.  If  there  be  three  unknown  quantities,  and  onlv 
two  equations  for  determining  them,  as 

ax'\'by-{-cz  —  df  and  ex-t-fy-{-gz=^h 
exterminate  one  of  these  quantities  in  the  usual   way,  and 
find  the  values  of  the  other  two  from  the  resulting  equa- 
tion, as  before. 

Then,  if  the  values,  thus  found,  be  separately  substi- 
tuted, in  either  of  the  given  equations,  the  corresponding 
values  of  the  remaining  quantities  will  likewise  be  deter- 
mined :  thus, 

6.  Let  there  be  given  x — 2y-\-z=b,  and  2x'{-y—z=.7f 
to  find  the  values  of  a*,  y,  and  z. 

Here,  by  multiplying  the  first  of  these  equations  by  2, 
and  subtracting  from  the  second  the  product  we  shall  have 

3z'-5y=dj  orz=     '     ^  =  l+y-{-±=wh.  ; 
o  > 

And  consequently  -^,  or  -^ — -^:=±=wh,  =^p, 
o  '5        3       3 

Whence  2/=3/). 

And,  by  taking  p=l,  2,  3,  4,   &c.  we  shall  have  y=3j 

6,  9,  12,  15,  &c.  and  2-=^6,  1 1,  )6,  2l,  26,  &c. 

But  from  the  first  of  the  two  given  equations 

x=b-\-2y^z; 

whence,   by   substituting  the  above  values  for  y  and  z,  the 

results  will  give  x=o,  ti,  7,  8,  9,  &c. 

And  therefore  the  first  six  values  of  x,  y,  and  z,  are  as 

below  : 


x=5 

6 

7 

8 

9 

10 

y=S 

6 

9 

12 

15 

18 

ze 

11 

16 

21 

26 

31 

Where  the  law  by  which  they  can  be  continued   is  suffi 
ciently  obvious. 


EXAMPLES  FOR  PRACTICE. 


1.  Given  3x=8!/-  16,  to  find  the  least  values  of  x  and 
y  in  whole  numbers.  Ans.  a:=8,  y~^ 


INDETERMINATE  ANALYSIS.  197 

2.  Given  \4x=5y-[-7y  to  find  the  least  values  of  x  and 
y  in  whole  numbers.  Ans.  a;=3,  y=7. 

3.  Given  ^7a;=16()0—  \6y,  to  find  the  least  values  of  a; 
and  y  in  whole  numbers.  Ans.  x=48,  i/=19. 

4.  It  is  required  to  divide  100  into  two  such  parts,  that 
one  of  them  may  be  divisible  by  7,  and  the  other  by  11. 

Ans.  The  only  parts  are  66  and  44., 

5.  Given  9x'-|-13?/=2u00,  to  find  the  greatest  value  of  a: 
and  the  least  value  of  y  in  whole  numbers. 

Ans.  x=215,  y=5f 

6.  Given  11x-l-5y=254,  to  find  all  the  possible  values 
of  X  and  y  in  whole  numbers. 

Ans.  .T  =  19,  14,  9,  4  ;  y—9,  20,  31,  42. 

7.  Given  17x-f  19i/-f-21z=40O,  to  find  all  the  answers 
in  whole  numbers  which  the  question  admits  of. 

Ans.   10  different  answers. 
S.  Given  5x-|-7?/4-  H^— 224,  to  find  all  the  possible  va- 
lues of  :r,  y,  and  z,  in  whole  positive  numbers. 

Ans.  The  number  of  answers  is  59. 

9.  It  is  required  to  find  in  how  many  different  ways  it  is 
possible  to  pay  20/.  in  half-guineas  and  half-crowns,  with- 
out using  any  other  sort  of  coin  ? 

Ans.  7  different  ways. 

10.  I  owe  my  friend  a  shilling,  and  have  nothing  about 
me  but  guineas,  and  he  has  nothing  but  louis  d'ors  ;  ho\y 
must  I  contrive  to  acquit  myself  of  the  debt,  the  louis  being 
valued  at  i7s.  a  piece,  and  the  guineas  at  "lis.  ? 

Ans.  I  must  give  him  13  guineas,  and  he  must 

give  me  16  louis. 

!  1.  How  many  gallons  of  British  spirits,  at  126-.,  13s,, 

and  IS.?,  a  gallon,  must  a  rectifier  of  compounds  take  to 

make  a  mixture  of  iOuO  gallons,  that  shall  be  worth  I7i>\  a 

gallon  ? 

Ans.   1  Hi,  at  125.,  1111  at  155.,  and  777^  at  18i\ 

PROBLEM  II. 

To  find  such  a  whole  number,  as  being  divided  by  other 
given  numbers,  shall  leave  given  remainders. 
s2 


198  INDETERMINATE  ANALYSIS. 

RULE. 

1.  Call  the  number  that  is  to  be  determined  x,  the  num 
bers  by  which  it  is  to  be  divided  a,  b,  c,  &c.  and  the  giveri 
remainders  /,  g,  h,  &c. 

2.  Subtract  each  of  the  remainders  from  x,  and  divide 
the  differences  by  a,  6,  c,  &c.  and  there  will  arise 

a-—/   a--g   x—h 


a 


,  &c.  =  whole  numbers. 


X f 

3.  Put  the  first  of  these  fractions  — ^-=0  and    substi- 

o 

tute  the  value  of  x,  as  found  in  terms  of  p,  from  this  equa 
tion,  in  the  place  of  x  in  the  second  fraction. 

4.  Find  the  least  value  of  p  in  this  second  fraction,  b} 
the  last  problem,  which  put  —r,  and  substitute  the  valup 
of  Xf  as  found  in  terms  of  r,  in  the  place  of  x  in  the  third 
fraction. 

Find,  in  like  manner,  the  least  value  of  r,  in  this  third 
fraction,  which  put  =«,  and  substitute  the  value  of  .r,  a.^ 
found  in  terms  of  s,  in  the  fourth  fraction  as  before. 

Proceed  in  the  same  way  with  the  next  following  frac- 
tion, and  so  on,  to  the  last  ;  when  the  value  of  x,  thus  de- 
termined, will  give  the  whole  number  required. 

EXAMPLES. 

1.  It  is  required  to  find  the  least  whole   number,  which, 
being  divided  by  17,   shall   leave  a  remainder  of  7,  an*' 
when  divided  by  26,  shall  leave  a  remainder  of  13. 
Let  x  =  the  number  required. 

Then  — r^,—  and  — — — =  whole  numbers. 

X 7 

And  putting  -— -=p,  we  shall  have  x=\7p-{'7. 

Which  value  of  x,  being  substituted  in  the  second  frac,- 

l7p+7—13     l7p-6 
tion,  gives =— ^_=t)t-/i. 

But  it  is  obvious  that  — -  is  also  =zvh. 
Jo 


INDETERMINATE  ANALYSIS.  199 

'26p     17D-6     9»-f6 
And  consequently  -^ 1-— =r-^^  =zwh, 

r>-l-  t  ft 

Where,  by  rejecting  ;j,  there  remains  —^ — wh.=r. 

26 

Therefore  p  =  ^6r-  18  ; 
Whence,  if  r  be  taken  =-1,  we  shall  have  p=^S, 
And   consequently,   a:=  l7/)  +  7=  17X8+7=  143  ;    the 
number  sought. 

2.  It  is  required  to  find  the  least  whole  number,  which« 
being  divided  by  11,  19,  and  29,  shall  leave  the  remain- 
ders 3,  5,  and  10  respectively. 

Let  x==-  the  number  required. 

Then  ,  and  —  whole  numbers. 

11         19  29 

X 3 

And,  putting  — 7-=Pi  we  shall  have  x=l  3;?-f-3. 

Which  value  of  x,  being  substituted  in  the  second^  frac- 

•       lip -2        , 
uon,  gives  —— — ~wh. 

Or  -^-X2=-^^-=;.^-^-=t.7^ 

\nd,  by  rejecting;?,  there  will  remain-— =^uih. 

Also  by  multiplication -Aj-^—X 6=—^ — ——£-—  —  ] 
ly  ly  ly 

~wh.  ; 

Or,  by  rejecting  the  1,  — — — =W'7<. 

But  -^-  is  likewise  =a;/i. 
19 

Whence  -r--— — ? — ^~^-— -=«'/».,  which  put  —r. 

Then  we  shall  h^ve 
p=19r~5,  and  a:  =  ll(19r-"5)4-3=209r-62. 
An(J  if  this  value  be  substituted  fgr  y  in  thQ  third  frac- 


20d  INDETERMINATE  ANALYSIS. 

(ion,  thero  will  arise 

209r-62     ^       ^     fir-4 

29  ^29  * 

Or,  by  neglecting  7r-2,  we  shall  have  the  remainiog 

.    .  .       6r— 4 

part  of  the  expression  —^-x-  ='»'/?.  ; 

But  by  multiplication, 

6r-4     ^     30r-20         .  r- 20 

X5= =r-\ -— =a)/i. 

29  29  29 

,. 20 

Or,  by  rejecting  r,  there  will  remain  —  ~-=a;/i.    which 

put  =s. 

Then  r=29s-}-20  ;  where  by  taking  5=0,  we  shall 
have  r=20. 

And  consequently 
a;— 209r~62=2U9X20-.62  =  4128, 
the  number  required. 

3.  To  find  a  number,  which  being  divided  by  6,  shalj 
leave  the  remainder  2,  and  when  divided  by  13,  shall 
leave  the  remainder  3.  Ans.  68, 

4.  It  is  required  to  find  a  number,  which  being  divided 
by  7,  shall  leave  5  for  a  remainder,  and  if  divided  by  9^ 
the  remainder  shall  be  2.  Ans.   110. 

5.  It  is  required  to  find  the  least  whole  number,  which, 
being  divided  by  39,  shall  leave  the  remainder  16,  and 
when  divided  by  56,  the  remainder  shall  be  27. 

Ans.   1147. 

6.  It  is  required  to  find  the  least  whole  number,  which, 
being  divided  by  7,  8,  and  9,  respectively,  shall  leave  the 
remainders  5,  7,  and  8.  Ans.  215, 

7.  It  is  required  to  find  the  least  whole  number,  which, 
being  divided  by  each  of  the  nine  digits,  1,  2,  3,  4,  5,  6,  7, 
C,  9,  shall  leave  no  remainders.  Ans.  2520. 

8.  A  person  receiving  a  box  of  oranges  observed,  that., 
when  he  told  them  out  by  2,  3,  4,  5,  and  6  at  a  time,  he 
had  none  remaining  ;  but  when  he  told  them  out  by  7  at 
a  time,  there  remains j  5  ;  how  many  oranges  were  there 
in  the  box  ?  Ans.  180 


DIOPHANTINE  ANALYSIS.  201 

OF  THE 

DIOPHANTINE  ANALYSIS. 

This  branch  of  Algebra,  which  is  so  called  from  its  in^ 
ventor,  Diophantus,  a  Greek  mathematician  of  Alexandria 
in  Egypt,  who  flourished  in  or  about  the  third  century  after 
Christ,  relates  chiefly  to  the  finding  of  square  and  cube 
numbers,  or  to  the  rendering  certain  compound  expres- 
sions free  from  surds  :  the  method  of  doing  which  is  by 
making  such  substitutions  for  the  unknown  quantity,  as 
will  reduce  the  resulting  equation  to  a  simple  one,  and 
then  findmg  the  value  of  that  (juaniity  in  terms  of  the  rest. 

It  is  to  be  observed,  however,  that  questions  of  this 
kind  do  not  always  admit  of  answers  m  rational  numbers, 
and  that,  when  tiiey  are  resolvable  in  this  way,  no  rule 
can  be  given  that  will  apply  in  all  the  cases  that  may  occur  ; 
but  as  far  as  respects  a  pa  ticular  class  of  these  problems 
relating  to  squares,  they  rnay  generally  be  determined  by 
means  of  some  of  the  rules  derived  from  the  following 
formulae. 

PROBLEM  I.  ' 

To  find  such  values  of  x  as  will  make  y/(ax^'\'bx'\-c) 
rational,  or  ax^-\-bx-{-c=  a  square.* 

RUI  E. 

1.  When  the  first  term  of  the  formula  is  wanting,  or  a 
=0,  put  the    side   of  the  square    sought  =n  ;  then  bx-^-c 


And,  consequently,   by  transposing  c,   and   dividing  by 
3  coefficient  6,  we  shall  have  x 
be  any  number  taken  at  pleasure. 


n"  —  c 
the  coefficient  6,  we  shall  have  a;= — r— ;    where   n    may 


*  The  coefficients  a,  6,  of  the  unknown  quantities,  as  well  as  the  absolute 
term  C,  are  here  suppos^^d  to  be  all  integers  .  (or  if  they  were  fractions, 
they  could  be  readily  reduced  to  a  common  square  denominator ;  which,  be* 
ing  afterwards  rejected,  will  not  alter  the  nnture  of  the  question;  since  any 
square  number,  when  multiplied  or  divided  by  a  square  number,  is  still  & 
square. 


202  DiOPHANTlNE  ANALYSIS. 

2.  When  the  last  term  is  wanting,  or  c=0,  put  the  side 
of  the  square  sought  =nx,  or,  for  the  sake  of  greater  gene- 
rality, =  — ;  then,  in  this  case,  we  shall  have  ar'+6a;~ 

And,  consequently,  by  multiplying  by  n^,   and  dividing 

A    2 

hy   Xy  there  will  arise   anrx-{-bn^=m^x.    and  x  =-^ r. 

m^  —  an'^ 

where  m  and  n,  both  in  this  and  the  following  cases,  may 

be  any  whole   numbers  whatever,    that  will  give  positive 

answers.* 

3.  When  the  coefficient  a,  of  the  first  term,  is  a  square 

number,  put   it  =<^^   and  assume   the  side  of  the  square 

L.        ,     .  ^'^       .  ,o      .  ,      .  ...  o  .  2dm     .  m^ 

sought  =6/x+  '  ;   then,   d^x--^bx-\-c  =d^x^-\ x-\ , 

n  n  n3 

And,  consequently,    by  cancelling  d'^r^^  and  multiplying 

hy  n'^   we   shall   have   bn'x-\-cn'^=2dmnx+m^j  and   a;= 

bn^ — 2dmn 

4.  When  the  last  term    c  is  a  square  number,  put  it  = 

e^,  and  assume   the  side  of  the    square  sought  =  — -fc  ; 

then,  ax^-i-bx  +  e^  =  —  ■■'{- x+e^.      And  consequent- 

n^         n  ^ 

ly,  by  cancelling  c^,  and  dividing  by  x,    we  shall  have  ax-r- 
,      m^x     2eni  bn^  —  2ei"n 

*=— ~-l and  x= — —. 

n^         n  m-^  -  an" 

5.  When  the  given  formula,  or  general  expression, 

ax^-hbx-rc 
can  be  divided  into  two  factors  of  the  form/x-f-g  and  hx-^- 


*  The  unknown  quantUy  5,  in  this  case,  can  always  be  found  in  integer? 
when  6  is  positive  ;  and,  in  Case  4  next  rollowin<j,  its  integral  value  can  al 
ways  be  determined,  whether  h  be  positive  or  negative.  See  Vol.  II.  of  Bon 
jiycastle's  Treatise  on  Algebra,  Art.  (HV 


DIOPHANTINE  ANALYSIS.  203 

:,  which  it  always  can  when  6-~4ac  is  a  square,  let  there 
)e  taken  {fx-\-g)X{hx'\-k)  =  —  {fx+gy;  then,   by  re- 

luction,  we  shall  have  x=^—- — -^r-—  ;   where  it   may   be 

)bserved,  that  if  the  square  root  of  6^  — 4ac,  when  rational, 
)e  put  =5,  the  two  factors  above  mentioned,  will  be 

axA ,  and  cH *, 

2  2a 

And,  consequently,  by  substituting  them   in  the  place  of 

he  former,  we  shall  have 

2a[a^  —am^) 

6.  When  the  formula,  last  mentioned,  can  be  separated 
nto  two  parts,  one  of  which  is  a  square,  and  the  other  the 
)roduct  of  two  factors,  its  solution  may  be  obtained  by 
)utting  the  sum  of  the  square  and  the  product  so  formed, 

til 

jqual   to  the  square  of  the  sum  of  its  roots,  and  —  times 

n 

)ne  of  the  factors,  and  then  finding  the  values  of  x  as  in 

he  former  instances. 

7.  These  being  all  the  cases  of  the  general  formula  that 
ire  resolvable  by  any  direct  rule,  it  only  remains  to  ob- 
serve, that,  either  in  these,  or  other  instances  of  a  differ- 
3nt  kind,  if  we  can  find,  by  trials,  any  one  simple  value  of 
he  unknown  quantity  which  satisfies  the  condition  of  the 
question,  an  expression  may  be  derived  from  this  that  will 
Jurnish  as  many  other  values  of  it  as  we  please. 

Thus,  let  p,  in  the  given  formula  ax^-\-hx-\-c,  be  a  value 
of  X  so  found,  and  make  a}r-{-bp'^c=q^. 

*  These  factors  are  found  by  putting  the  given  formulae  aa?3-|-6x+cc= 
[>,  and  then  determining  its  roots :  which,  by  the  rule  for  quadratics,  are 

Whence,  if  b2  — 4ac  be  a  square,  of  which  the  root  is  5,  we  shall  have  a,  x-\m 

. --,  and  a:+;r-  +  r-»  for  the  divisors  of  ax2  -^fer-f-c,  or  «a!-f-— r— . 

•2a    2a  2a      2a  2 

nnd  x4'--~~,  for  its  two  factors,  as  in  the  above  rule. 


204  DIOPHANTINE  ANALYSIS.  i 

Then,  by  putting  x=-y-\-p,  we  shall  have  ax^+^ar-j-cr^ 
a{y-\-pY+b{y-\-p)-^c=ay2-^{2ap+b)y-\-ap^-\-bp+c,  oi 
ax^'{-bx-\-c=^ay^'i'  (2ap-f-6)?/-}-9^ 

From  which  latter  express-ion  the  values  of  ?/,  and  con* 
sequently  those  of  x,  may  be  tound,  as  in  Case  4. 

Or,  because  c=/ — bp  —  ap"^,  if  this  value  be  substitut- 
ed  for  c,  m  the  original  formula  ox^-f  6i4-Cj  it  will  become 
a{x'^—p'^)-^b[x—p)-\-q",  or 

9^+(a;-/?)X(ox-f-op+6)—  a  square  ; 
v/nich  last  expression  can  be  resolved  by  Case  ri. 

It  may  here,   also,  be   farther   observed   that  by  putting 

2  L 

the  given  formula  ax--\-bi-^c  =  —  ^  and  taking   3c=— - —  ; 

we  shall  have,  by  substituting  this  value  for  x  in  the  former 
of  these  expressions,  and  then  nmltipivmji  by  4a,  and  trans- 
posing the  terms  r/iy--l-(6 — 4ac)=^^^  ;  or,  putting,  for  the 
sake  of  greater  simplicity,  6^~4ac=6'.  this  last  expression 
may  then  be  exhibited  under  the  form  t7t/^+6=2'^?  where 
it  is  obvious,  that  if  01/^4- (^^-4tfc),  or  its  equal  ay^-{-b', 
can  be  made  a  square,  ax'^+bx-\-Cy  will  also  be  a  square. 

And  as  the  proposed  formula  can  always  be  reduced  to 
one  of  this  kind,  which  consists  only  of  two  terms,  the 
possibility  or  impossibility  of  resolving  the  question,  in 
this  state  of  it,  can  be  more  easily  perceived.* 

EXAMPLES. 

1.  It  is  required  to  find  a  number,  such  that  if  it  be 
multiplied  by  5,  and  then  added  to  19,  the  result  shall  be 
a  square. 

Let  x=  the  required  number ;  then,  as  in  Case  1,  hx-\" 

*  It  may  heie  be  observed,  that  an  infinite  number  of  expression?,  of  the 
kind  fl?/'^  •+ (6** — oc),  or  ai^"-4_6'c=2:^,  here  mentioned,  are  wholly  irre- 
solvable ;  among  which  we  may  reckon 

2/  3:  3,  5y^  It  6,  7y^  Hh5,  &c. 
none  ef  which  can  ever  become  squares,  whatever  number,  either  whole  or 
fractional,  be  substituted  for  y\  although  there  are  a  variety  of  instances  in 
which  the  value  of  y  may  be  found,  even  in  integers,  so  as  to  render  the  for- 
mula 01/2  4_  6  =  2^^ 

For  a  further  detail  of  which  circumstance,  as  well  as  for  other  particulars 
s-elating  to  this  part  of  the  subject,  see  the  second  volume  of  Eider^s  Algebra. 
or  the  second  rolume  of  Bonnyccistle^ ?  .QJgtbrn. 


BIOPHANTINE  ANALYSIS.  205 

-  .       o  n2  — 19        ,         .    .        ., 

i:B=ii%  or  a;= ;  where  it  is  evident  that  n  may  be 

any  number  whatever  greater  than  ^^  19. 

Whence,  if  /*  be  taken  -  5,  6,  7,  respectively,  we  shall 

25-19  36-19     „  49  —  19     ^ 

nave  x= — -  -  =li,  or  — ~ — =3f ,  or  — - — =6  ; 

the  latter  of  which  is  the  least  value  of  x,  in  whole  num- 
bers, that  will  answer  the  conditions  of  the  question  ;  and 
«onsequenily  oa:-l- (9±=6X6-h  i9=3o-f  19=49,  a  square 
number  as  w;is  required. 

2.  It  is  required  to  find  an  integral  number,  such  that  it 
shall  be  both  a  triangular  number  and  a  square. 

It  is  here  to  be  observed,  that  ail  triangular  numbers  are 

of  the  form  — --^  ;  and  therefore  the  question  is  reduced 
■Z 

x^A-x                          2a:"+2x 
to  the  making  — - — ,  or  its  equal —  a  square. 

Where,  since  the  divisor  4  is  a  square  number,  it  is  the 
same  as  if  it  were  required  to  make  2x^+2x  a  square. 

(Ifi-X  \  tfl^X^ 

—  I  ^— — —,  agreeably  to  the 

method  laid  down  in  Case  2. 

Then,  by  dividing  by  x,  and  multiplying  the  result  by 
n\  the   equation  will   become    2n^x+2n^'^m^Xj  or  (m^-^ 

2n^)x=2n^ ;  and  consequently  a^=— a — q~2  »  where,  if  n 
be  taken  =2,  and  m-=3,  we  shall  have  x=8,  and  — -— 

fi4-l— ft      72 

= — =36,  which  is  the  least  integral   triangular 


2     ~2 

■umber  that  is  at  the  same  time  a  square. 

3.  It  is  required  to  find  the  least  integral  number,  such 
that  if  4  times  its  square  be  added  to  29,  the  result  shall 
be  a  square. 

Here  it  is  evident,  that  this  is  the  same  as  to  make  4x^ 
4-29  a  square. 

And,  as  the  first  term  in  the  expression  is  a  square,  let 

T 


29§  DIOPHANTINE  ANALYSIS. 

4x^4-29=  (2x'{'-y^4.x^+—x+—, ;  agreeably  to  Cast 

3. 

_,         4m     ,  m^     ,^^        4m       ^^     m^ 

Then,  -    a;H — ^=29,  or  —  a:=29  -  — -;     and      conse- 

'29n^—  m^ 
quently  x=- — ;  where,  if  m  and  n  be   each  taken 

29 J 

^1,  we  shall  have  a:= =7,  and   4x^-\-t^=^^y.AQ-{' 

29=2^5= (15)2,    ^hich  is   a  square  number,  as   was  re- 
quired. 

4.  It  is  required  to  find  such  a  value  of  x  as  will  make 
7ac" — bx-\- 1  a  square. 

Here  the  last  term  1  being  a  square,  let  there  be  taken., 
according  to  Case  4, 

/fn  V  „     m^  „      2m     . 

7x2-5.T+l  =  r-3;-n"=-i-T' x+l. 

^n  ^       n^  n 

Then,  by  rejecting  the  1  on  each  side  of  the  equation. 

77 1^  '27 ft 

and  dividing  by  x,  we  shall  have   Ix — b=—^x .    and 

n^         n 

consequently  x=- — —  ttt  '■>  ^^^ere,   if  r/i  and  n  be  each 

2  --  5     3 
taken  =1,  the  result  will   give  a:= 7~r~2J  ^^  ^y  ^ak- 

43 45 

ing  n=3,  and  m=-'d^  we  shall  have  x=^— — r-=3,    which 

64  —  63 

makes  7X32—5X3+1  —  19=72,  as  required. 

5.  It  is  required  to  find  such  a  value  of  x  as  will  make 
8a;2-r  14j4-<^  a  square. 

Here,   by   comparing   this  expression  with  the  general 
formula  ax--^hx-\-c^  we  shall  have  a  =  8,  6=14,  and  t'  =  6. 

And,  as  neither  a  nor  c,  in  the  present  instance,  are 
squares,  but  /^^ -.4ac=196  — 19"=4  is  a  square,  the  given 
expression  can  be  resolved,  by  Case  5,  into  the  two  follow 
ing  factors  8x-|-ti,  and.T-f-l. 


DIOPHANTINE  ANALYSIS.  201 


Let,  therefore,  Sx^-^\4x+6={8x-^Q){x-\'\)=—J^x-{- 

I  Yf  agreeably  to  the  rule  there  laid  down. 
Then  there   will  arise,   by   dividing  each  side  by  a;-}"  1? 

n 
And,  consequently,  by  multiplication  and  reduction,  we 

shall  have,  in  this  case,  .T= —J j;    where   it    appears, 

that,  in  order  to  obtain  a  rational  answer,  -5-  must  be  less 

than  8,  and  greater  than  6. 

Whence,  by  taking  m=5,  and  n=2,  we  shall  have  1;  = 
25-24     1       ^.  .         ,        ^,14,^     400      .20.3 

quired. 

6.  It  is  required  to  find  such  a  value  of  a;  as  will  make 
2x^  —  2  a  square. 

Here,  by  comparing  this  with  the  general  formula  ax^ 
-{-6r-4-c,  as  before,  we  shall  have  a  =  2,  6-=o,  and  r=z  —2, 

And,  as  neither  «  nor  c  are  squares,  but  b^ — 4ac=— 4cfc 
=  -4(2  X  — 2)  =  iH  is  a  square,  the  root  of  which  is  4, 
the  given  expression  can  be  resolved,  by  Case  5,  into  the 
two  factors  2r— 2,  and  a-f-1,  or  2(a?— 1),  and  (x-j-l), 
which  is  evident  indeed,  m  this  case,  from  inspecuon. 

Let,  therefore,  2a:'-  -2=  2(a;  -  1)  X(a:-h  l)=—  (   +1)*, 

agreeably    to    the   rule  ;  and   there    will  arise  by  division 

2x  —2^=  — ;;(.c4-  !)•      And,  consequently,  by  multiplication, 

and  reducing  the  result,  we  shall  have  x=  —  ^r—  -;  where, 

by  taking  «  =  !,  and  'n  — 1,  we  shall  have  a;=3,  and  2a;^  — 
2 -IS  — •£=l»i  =  (i)%  or  taking  n=2,  and  m—S,  there- 
suit  will  give  a;=  —  17. 

But  as  X  enters  the  problem  only  in  its  second  power, 
-}-17  may  be  taken  insiead  of  — 17  ;  since  either  of  them 
gJve2.T'~~2=576  =  (24/. 


208  BIOPHANTINE  ANALYSIS, 

7.  It  is  required  to  find  such  a  value  of  x  as  will  make* 

6x^'{-^^6x-\-'3  a  square. 

Here,  by  comparing  the  expression  with  the  general 
formula,  we  shall  have  a=5,  6  --  ib,  and  c=-7. 

And  as  neither  a  nor  c  are  squares,  but  6^ — 4ac=l296 
— 140— 115b=(34)^,  IS  a  ^;quare,  it  can  be  resolved,  as  in 
the  last  example,  into  the  two  factoKs  oj-hl,  and  x+7. 

Whence,   putting   53:2-i-36r-i-7  =  (5a-f  1  jX(a:4'7)=— 

11 

{x-^iy,    there  will   arise,  by    dividing  by  x-f7,  5a:-i-l  = 

And,  consequently,   by  multiplication,  and  reducing  the 

resulting   expression,   we   shall    have  x— — '  where, 

taking  m=2,   and   n  =  l,   the   substitution    will  give  a;  = 
7  X4  —  1 

-=-— — j=27,  which  makes  6 X (27)^+^6X27  +  7-4624 
D  X  I  —  4 

=  (68)^,  as  required. 

8.  It  is  required  to  find  such  a  value  of  x  as  will  make 
6.T^+13x-}-  lU  a  square. 

Here,  by  comparing  the  given  expression  with  the  ge- 
neral formula  aa;-+6a;4-c,  we  I'ave  a=6.  6=^1:-},  and  c=: 
10.  And  as  neither  a,  c,  nor  b''-  4i/c,  are  squares  the 
question,  if  possible,  can  only  be  resolved  by  the  method 
pomted  out  in  Case  6. 

In   order,    therefore,  to   try  it   in  this   way,  let  the  first 
simple  square  4,  be  subtracted  from  it,  and  there  will  re 
main,  in  that  case,  bx-+  iSx-f-^;. 

Then,  since  (13j2- 4(6  X0)  =  1  9-144=25,  is  now  a 
square,  this  part  of  the  formula  can  be  resolved  by  Case 
S,  into  the  two  factors  ; 

3x-\-2f  and2r+3. 

Whence,  by  assuming,  according  to  the  rule,  Sx^-f-lSat 

+  l0=4-|-(3x-{-2)  X(2a;-f3)=  ^  2-f  ^  (3a;-f2)  ^  ^=4^ 
—(3a?-l-2)-}— J  (3x4-2)3,  we  shall  have,   by  cance!Un|- 


DIOPHANTINE  ANALYSIS.  209 

the  4  on  each  side,  and  dividing  by  3x-{-2 ;  2ic-l-3= — 

n 

+-(3x4-2). 

And,  consequently,  by  multiplying  by  n^,  and  transpos- 
ing the    terms,  we   have  2n^x — 3m^.T=4mn+2m^ — 3n^,  or 
___4/Am-f2/n2— 3«2 

Where  putting  m=3,  and  r.=3,  the  result  will  give  a:= 

24+8-27     5        ..      .     ,  ,  ,..        .        ,^  ,    „ 

— TTj — r7~~c'  ^*"  "  ^^  "®  tai-jen  =13,  and  /j=l7,  we  shall 

4Xl7Xl3+2X(i    )^—3X(17)2     355      ^ 

nave  x= —  — i— 1.= =5. 

2(17)^— :^;1H  ^'  71 

Which  makes  6 X (5/+ i 3X5+ 10=225=,  (15)2,  as 
required. 

9.  It  is  requirt^d  to  find  such  a  value  of  a;  as  will  make 
13a;^+15a'+7  a  square. 

Here,  by  connparuig  this  with  the  general  formula,  as 
before,  we  have  a=  I  ,  6  =  15,  and  c  ~  7.  And  as  neither 
a,  6,  nor  6^ — 4ac,  are  squares,  the  answer  to  the  question, 
if  it  be  resolvable,  can  ()r)ly  be  obtained  by  Case  6.  la 
order,  therefore,  to  try  it  in  that  way,  let  (1 — xy  or  1— 
2x-}-x^  be  subtracted  from  the  given  expression,  and  there 
will  remain  l2x-+ 1  7j:-  +  6. 

And  as  ( 17  )-—  4(6  X  l  2),  which  is  =  I ,  is  now  a  square, 
this  part  of  the  formula  can  be  resolved  by  Case  5,  into 
the  two  factors  4x+3  and  3z-l-2.     Whence,  assuming  13 

:c2+15x  +  7=(l— xf+(4.x+3)X(3x+2)-r  |  (l  —a:)  + 

^'(3x-+2)  I  2=(l_a-)^+^(l— x)X(3x4-^)+^(3a;  + 

2)^,  we  shall  have,  by  cancelling  (1— x)-,  and  dividing  by 

3x+2  ;  4x+3=—  1 — x)  +  '"(3x+2) ;  aud,  consequently, 

by  multiplying  by  71^,   and  transposing  the  terms,  there 
will   arise   4n^x+2mnx  — 3r/*^x=2mn+2w^ — 3n^,   or   x~ 

4n^+2rnn— 3m2*  T  2 


2id  DIOPHANTINE  ANALYSIS, 

Where  putting  m  and  n  each  =1,  we  shall  have  x=^ 
|g=^=i,  which  makes  i:+^+7=^+:^+^=l|l 


=  J  —  j    ,  as  required. 


10.  It  is  required  to  find  such  a  value  of  x  as  will  make 
7a:^+"2  a  square 

Here  it  is  easy  to  perceive  that  neither  of  the  former 
rules  will  apply. 

But  B.ii  the  expression  evidently  becomes  a  square  whet 
ar=l,  let,  therefore,  a  =  l  -f-y,  according  to  Case?,  and  we 
shall  have 

Or,  putting  9+14?/+72/^=(3-h-i/)^  according  to  the  rule, 
and  squaring   the   right   hand   side,    9-\'J4y -\- Ty^^dA- 

— 2/+-rr- 
n        n 

Hence  rejecting  the  9's  and  dividing  the  remaining  terms 

by  ?/,  we  have  ln^y-\- 1  ^n^=Qmn-\-in^y  ;  and,  consequent- 

Qinn — XAnr  ,  ,   ,  Qiim — 14n^  , 

^y>  y'^-iTT --'  ^"°  ^=1"T— ;^-T r-5    where   it  is 

evident  that  m  and  n  may  be  any  positive  or  negative  num- 
bers whatever. 

If,  for  instance,  m  and  n  be  each   taken   =1,  we   shall 

4  1 

have  y=  —  r  and  x——-.     Or,  since  the  second  power  o* 
3  o 

X  only  enters  the  formula,  we  may  take,  as  in  a  former  in- 

stancej  a:=i,  which   value  makes  7a;^-f'^=i^+2=|H-y 

=2_5  a  square. 

Or,  if  m=3  and  n= — 1,  we  shall  have  a;  =  17,  and  7// 
-1-2=7  X(17)2-f2=2025=(45)2,  a  square  as  before. 

And  by  proceeding  in  this  manner,  we  may  obtain  as 
many  other  values  of  x  as  we  please. 

PROBLEM  II. 

To  find  such  values  of  x  as  will  make  ^{ax^'{'b3i^+c 


DIOPHANTINE  ANALYSIS,  2U 

jc-^d)  rational,  or  ax^-^-bx^-^cx-h'i^  a  square.  This 
problem  is  much  more  limited  and  difficult  to  be  resolved, 
than  the  former  ;  as  then.'  are  but  a  few  cases  of  it  that 
admit  of  answers  in  ratumal  numbers  ;  and  in  tnese  the 
rules  for  obtaining  them  are  of  a  very  confined  nature  j 
being  mostly  such  as  are,  subject  to  certain  limitations,  or 
that  admit  only  of  a  few  simple  answers,  which,  in  the  in« 
stances  here  mentioned,  may  be  found  as  followa. 

KULE. 

1.  When  the  third  and  fourth  terms  of  the  formula  are 
wanting,  or  c  and  d  are  each  =0,  put  the  side  of  the  square 
sought  =nXf  then  ax  -t-bx^=n^x^. 

And,  consequently,  by  dividing  each  side   of  the  equa^ 

tion   by  x-,  we  shall   have  ax-\-b=ti^y  or  x~ ,  wherf 

•'  a 

n  may  be  any  integral  or  fractional  number  whatever. 

2.  When  the  last  term  (i  is  a  square,  put  it  =c^,  and  as- 

d 
same  the  side  of  the  required  square  =e+— ^' »    and  th« 

reversed    formula    is    c^  -f  ca;  +  bx^-{-ax^—e^-{-cx-\'-—„x^. 

Whence,  by  expunging  the  terms  e^-\-cx,  which  are  comr 
m(Mi,  and  dividing  by  j-,  we  shall  have,   ^ae^X'{-^be'^B=c^ ; 

and,  consequently,  x= — 1~~* 

c        4be^ c^ 

Or,  if,  in  the  same  case,  there  be  put  «H"o~^"' —  gii — ^^ 

for  the  side  of  the  required  square,  we  shall  have,  by  squar- 

c(4be^^c^) 
ing,    e^  +  cx-\-  bx'  +  ax'^  e'-^cx+bx^-^  6e'  '^ 

(ibe^—c^Y 

1 —-a;*.     And   as   the   first   three  terms  {e^-^-cx-i- 

64  e^ 
hx^)   are   now   common,   there   will   arise,  by  expunging 
them,  and  then  multiplying  by  64c%   64aeV=Sce\46c2— 
c2)x2+(46e2_c2)V. 

Whence,  by  dividing  each  side  of  this  last  equation  by 
nPf  and  redvcing  the  result,  we  shall  have 


:2l^  DIOPHANTINE  ANALYSIS. 

_64a€«-  8ce^(46e-~c^) 

which  last  method  gives  a  new  value  of  a:,  different  from 
that  before  obtained. 

It  must  be  observed,  however,  that  each  of  these  forms 
fail,  when  ihe  second  and  third  terms  of  the  given  formu- 
la are  wanting,  or  6  and  c  '^ach  =0.-*= 

3.  When  neither  of  the  above  rules  can  be  applied  to 
the  question,  the  formula  can  be  resolved,  by  first  finding, 
by  trial,  as  in  the  former  problem,  some  value  of  the  un- 
known quantity  ttiat  makes  the  given  expression  a  square  : 
in  which  case  other  values  of  it  may  be  determined  from 
this,  when  they  are  possible,  as  follows  . 

Thus,  \etp  be  a  value  of  oc  so  found,  and  make 

Then,  by  putting  x=^y-\-p,  we  shall  have  ap^-^hp^-^cp 

.X.d^a{y+pY-\-tj{y\-pf^c{y+p)'{-d=^af-\-{^ap-itb)y^ 
+  (3/5'+2/?+c)i/H-ap^+6/?^-fc/?-fc/,orax^4-6x'4-cx-hc^= 

From  which  latter  form,  the  value  of  y,  and  consequent- 
ly that  of  a;,  may  be  found  by  either  of  the  methods  given 
in  Case  2. 

It  may  also  be  further  remarked,  that  if  the  given  for- 
mula, in  any  case  of  this  kind,  can  be  resolved  into  factors, 
Buch  that  one  of  them  shall  be  a  square,  it  will  be  suffi- 
cient to  make  the   remaining  factor  a   square,  in  order  to 

*  In  the  first  of  these  methods,  the  assumed  root,  e-i-  — a,  is   detcrrained 

'2c 
b}'  first  taking  it  in  the  form  e-f-nx,  and  then  equating  the  second  term  ot 
it,  when  squared,  with  the  corresponding  term  of  the  original  formula  ;  when 

c 
It  will  be  found,  that  n  =•—. 
2e 

In  like  manner  the  assumed  root  e-\~—xJ x2  in  the  second  ine- 

ihod,  is  determined  by  first  taking  it  in  the  form  e-^nx-\-mx2^  and  then 

equating  the  second  and  third  terms  of  it,  when  squared,  with  the  corres- 

c 
pending  terms  of  the  given  formula;  when  it  will  be   found,  that  n= — 

4ie2 — c2 
miS  7U  =  — . 

8^3 


DIOPHANTINE  ANALYSIS.  213 

render  the  whole  expression  so ;  since  a  square,  multipli. 
©d  or  divided  by  a  square,  is  still  a  square.* 

EXAMPLES. 

1 .  It  is  requh-ed  to  find  such  a  value  of  x  as  will  make 
Mx^-^-Sx^  a  square. 

Let  the  given  expression  llx^4-3a-^=n^a;^ :  agreeably 
to  Case  1. 

Then,  bv   dividing  by  ar-,  we  shall    have   I  \X'{-3=n^  ; 

and,  consequently,  .t=  -_—  ;  where  n  may  be  any  num- 
ber, positive,  or  negativ*^,  that  is  greater  than  v^3. 

Taking,   therefore,  n=2,  3,  4,   5,  &c.    respectively,  we 
1      fi     1  '^ 
shall  have,  in  this  case,  ^~"~;»  tt»  TT*   ^^  2,  the   last   of 

which  is  the  least  integral  answer  that  the  question  admits 
of. 

2.  It  is  required  to  find  such  values  of  x  as  will  make 
x^^2x^-\-2x-\-[  a  square. 

Here  the  last  term  i ,  being  a  sqtiare,  let  '  +2a; — 2x^4- 
x^ — {l4-xf~l-{-2x-{'X^,  agreeably  to  the  first  part  of 
Case  2. 

Then,  since  the  first  two  terms,  on  both  sides  of  the 
equation,   dt^stroy  each  other,  ^ve  shall   have  x^ — 2x^=a;^, 


*  The  method  of  determining  the  factors  of  which  any  formula  is  composr 
ed,  when  it  cat*  be  done,  is  to  put  the  i^jvf>n  expression  =  0,  atid  then  find 
the  roots  r,  v,  &,c.  of  the  equatioas<»  formed  ;  each  of  wliich  will  give  afac« 
tor  a: — r,x — r',  and  these  are  generailv  easily  discovered,  as  we  here  seek 
only  the  rational  roots,  which  are  alv7ay&  divisors  of  the  absolute  term,  or 
of  that  which  do.s  not  contain  x. 

Thus,  the  fornuda  x3 — x2 — x  +  I  is  \esoIvable  into  the  factors  (1 — x)  X 
(l-|-jr)X(l  -^),  or  (1— x)2X(l-f  a);  ^nd  bv  puttin}?  1 -|-T=9na,  we 
have  X  =s/»a— .1  ,  where,  if  n  be  taken  qual  to  any  number  whatever,  a;3— 
x2 — x-J-l  will  bf.  a  square  ;  though,  by  any  other  mode  of  solution  itwould 
be  difl5cult  to  find  even  two  or  three  values  of  x. 

It  may  here  also  be  observed,  there  ■\re  but  few  questions  in  this  problem 
that  can  be  determined  in  whole  numbers.  Several  of  them,  likewise,  ad» 
mit  only  of  one  answer,  and  others  art-  totally  irresolvable,  either  in  intefcere 
or  fractions.  Thus,  if  it  were  requiied  to  make  a; 3  4- 1  a  square,  the  only 
positive  value  of  x  that  renders  this  possible,  b  2 ;  and  the  raakiog  of  dX^ 
—1  a  square,  is  impossible, 


214  DIOPHANTINE  ANALYSIS. 

or  x^=-3x^,  and  consequently  a:=3  ;  which,  by  substitu- 
tion, makes  l-f-2aj  -  2x^-tx-'—  I  -}-6  « 18+27=  l6,asquare, 
as  required. 

Again,  by  putting  x=^y-\-S,  according  to  Case  3,  we  shall 
have  H-2x  2x^-1-3;^=  l+:£(i/+3)-2v2/+3)2-|-(y +3)^= 
le+lly  +  ly^-hf. 

And,   consequently,  by   making    16-{-172/+7^^4-2/^=(^ 

J  7  2H9 

-J— _y)2=i 6_}_j7y-j_ I__y2  agreeably  to  the  first  part  of 
8  64 

Case  2,  by  cancelling  \6+lly,  there  will  arise  ly^-^-lf^^ 

289  „  ,  ^     289 

-64-2/^ory+7=--. 

__..  289     ^     289—448  159        ,  „ 

Whence  y=— 7-  — = --,  and  x  •-=  3  — 

^      64  64  64 ' 

159      192—159     33    ,  .  ,         , 

— — = — ==r-»  lor  another  value  of  a;. 

64  64  64 

Which  number,  being  substituted  in  the  original  formula, 

42  w»2  656 

makes   \-{-2x—2x^-\'X^~ .   ==[ i^asquare,  asbe- 

2^;2t44      ^162  ^ 

fore. 

3.  It  is  required  to  find  such  values  of  x  as  will  make 
3x^-  5x^+6^-4-4  a  square. 

Here,  4  being  a  square,  let  4  +  6a:  — 5a:^+4x"'=(2+ 
^xY=4-{'6x-{-^r'\  as  in  the  first  part  of  Case  2. 

Then,  since  the  tirst  two  terms  on  each  side  of  the 
equation  destroy  each  other,  we  shall  have  3x^  -  5x^=f 
x^  or  3x — 5=|,  and,  consequently,  in  this  case  x  — 
y-l-^29 

3  12' 

29  29  45 

Whence  (2+1  X—)2—(2+--)2=(—;:2    a    Square,    as 

was  required. 

Or,  by  the  second  method  of  the  same  Case,  let  4+6x 

Q  ftT  ft/t  1 

^5x2+3x^=(2  +  fx-— a-^)-=4  +  6x-  5^'--pg  ^'  +  g^g 
r^ ;  then,  as  the  first  three  terms  on  each  side  of  this 


DIOPHANTINE  ANALYSIS.  215 

equation  destroy   each  other,   we  shall  have  — -a:''— -— x^ 

'^56        16 

84 1         ST 
=3.r^  or  o^rx-— =3,  or  84  irr— 1392=768  ;  and  con^ 

25b        16 

13-^2+768     2160       ,.,    . 
sequently,  a;= — — — — -=-—     ,  which  is  another  value 

of  a:,  that,  being  substituted  in  the  original  formula,  will 
make  it  a  square. 

4.  It  is  required  to  find  such  values  of  x  as  will  make 
jc^+S  a  square. 

Here,  it  is  evident,  that  the  expression  is  a  square  when 
x=^l.     Let  therefore   a:=l+2/)and  we   shall  have  3+x^ 

=44-32/+3r+f/- 

And  as  the  first  part  of  this  is  a  square,  make,  according 
to  the  first  part  of  Case  2,  4+:iy-fSy'+y'=[2+^yy= 
4+32/+T6  */^*  Then,  because  the  first  two  terms  on  each 
side  of  the  equation   destroy  each  other,  we  shall  have  y^ 

Whence  2/=-_3=-— =-_,  and  .^=1-^^- 

16-39         23       ,.  .   .  .      , 

— __ — =:  — -  ;  which  IS  a  second  value  of  x. 
Id  16 

3       39 

Again,  let  4'^Sy-h3y'+f--{2-\--y-h-yy=4-\'3y-\^ 

117         1521 
'    ^^^^'^TgS^^'^ioie  ^**   according   to    the   second   part   of 
Case  2. 

Then  as  the  first  three  terms  on  each  side  of  the  equa- 

1521         117 
tlon  destroy  each  other,  we  shall  have  —7-^2/''+ tt52/^  ~  if^ 

1521     ,  117 

«r u+ =  1. 

4096^^1^8 

wu  1  352         ,,  ,    352       1873      ,  .  ,  . 

Whence,  also,  ^= j^^,  and  0;=:  1  -|-^__=_^,  which  is 

a  third  value  of  a:. 

And  by  proceeding  in  the  same  way  with  either  of  these 
fiew  values  of  x  as  with  the   first,  other  values  of  it  may 


il6  DIOPHANTINE  ANALYSIS. 

he   obtained  ;  but  the  resulting  fraction  will  become  c«ii 
finually  more  complicated  in  each  operation. 

proplem  ni. 

To  find  such  values  of  x  as  will  make  v^(aa;''+6x^-|- 
KX^+dx-]re)  rational,  or  ax'^-\-bx^-\-cx'^-^(ix-\-€=  a 
square. 

The  resolution  of  expressions  of  this  kind,  in  which  the 
indeterminate,  or  unktiown  quantity,  rises  to  the  fourth 
power,  is  the  utmost  hrnit  of  the  researches  that  have  hi- 
therto been  made  on  tbrniulas  affected  by  the  sign  of  the 
square  root  ;  and  in  this  Problem,  as  well  as  in  that  last 
given,  there  are  only  a  few  particular  cases  that  admit  of 
answers  in  rational  numbers  ;  the  rest  being  either  im- 
possible, or  such  as  at?ord  one  or  two  simple  solutions ; 
which  may  generally  be  found  as  follows  :* 

RULE. 

1 .  When  the  last  term  c,  of  the  given  formula,  is  a  square., 
put    it    =/2,    and    make  f- -\-dx+cx^'jrbx^-\-ax'^=^(^f-\^ 

Then,  by  expunging  the  first  three  terms,   which  are 
common  to  each  side  of  the  equation,  there   will  remain 

€[uently,  by  dividing  by  x^  and  reducing  the  result,  we 
,.  1.  u  64hr-8df%4cp^cP) 

shall  have  at—  — ^2^^ — iFTi^  TTTfi —  5 

which  form  fails  when  the  coefficients  c  and  d,  or  h  and  (/, 
are  each  =0. 


*  As  an  instance  of  what  is  above  said,  it  may  be  observed  that  the  on! y 
value  of  a  that  renders  the  formula  2x4 — 3x2—2  a  square,  is  1 ;  and  the  for- 
-wula  a« — ya  -j-l,  can  nerer  be  a  square,  except  when  ccsa  -^1,  or  —1. 


DIOPHANTINE  ANALYSIS.  217 

2.  When  the  coefficient,  a,  of  the  (irsi  term  of  the  for" 
mula,  is  a    square,   put  it  =<.'^,  and    make  gV+^r'+cx^ 

Then,  c?r4-e=-^-— -x+-^- — -— -;     and    conse- 

Ucg^-h''Y—64eg'  .   ,     ^ 

<iuently,  x=  — ~Q — ^  ^r -^ — roT  ;  which  form  hkewise 

fails  under  similar  circumstances  with  the  former. 

3.  When  the  tirst  and  last  terms  of  the  formula  are  both 
squares,  put  a=<r'^,   and   6=/^,  and    make/^4-^.'c+cx^+ 

-^g'xK     Then  cx^--\-bx^={2fg  +i'-y^^x. 

And,  consequently,  x=-^- — -^^^ ,.      ,    * — . 

Or,  because^  enters  the  given  formula  only  in  its  se- 
cond power,  it  may  be  taken  either  negatively  or  positive- 

\y  ;  and,  consequently,  we  shall  have  x=^ — Tf,r\    f-^ — -^ 

So  that  this   mode  of  solution  furnishes  two  different  an- 
swers. 

Also,  if  there  be  taken  for   another   supposition/^-j-r/x 

-^cx'+  bx'-hg'x'  =  (/+l.x+gx^)^=/2+^x  +  {2fg  -h 

h'  ^  hf 

— x^'\-bx^-\-g^x'^i     hence    by   cancelling,    dx-\-cx^=—x 
4g2  g 

f  (2fe+£-.)^^ ;  and  consequently,  x^^^^^^j^^. 

And  because/  enters  the  given  formula  only  in  the  se- 
cond power,  it  may  be  taken  either  negatively  or  positively ; 

afld,  consequently,  we  shall  also  have  a;=  ...^    f/nr   t    x* 


218  DIOPHANTINE  ANALYSIS. 

So  that  this  solution  likewise  furnishes  two  values  of  x, 
which  are  each  different  from  the  former. 

but  these  tbrms  all  fail  under  similar  circumstances  with 
those  of  the  second  Case. 

4.  When  neither  the  first  nor  the  last  terms  are  squares, 
the  formula  cannot  be  resolved  in  any  other  way,  than  by 
first  endeavourijt^  to  discover  by  trials,  some  simple  value 
of  the  unknown  quantity,  that  will  answer  the  ronditions 
of  the  question  ;  and  then  finding  other  values  ol  it.  accord- 
ing to  the  methods  pointed  out  in  the  two  last  problems. 

Thus,  let  p  be  a   value  of  x  so  found,  and  n»ake    ap^-\' 

Then  by  putting  x=y+p,  we  shall  have  op^-^bp^-hcp^ 
'^dp-^e-^a{y-\-pY-\-b{ij-{-py-^c{y-j-pf+(i{rj-{-f)-{-e=ay'' 
'\-{ap-i-h)y'-l{bop--{-'6hp-\-c)y^-  {4op-  -^Sbfr  -^2cp  +  d)y■{' 
ap'^^\-bp' -i-cir-^-dp-Jre.  or  ax^-]-bx'^-^cx--\-dx't-€=ay'^'\-{ap 
+b)y^  +  ('  a/?^-i-  Sbp  +  c)y^^-{-{4ap''-{-2bp^-h'icpA-d)y'\-q\ 
From  which  last  formula  the  value  of  y,  and  consequently 
that  of  X,  may  be  found  by  Case  1. 

EXAMPIEH. 

1.  It  is  required  to  find  such  a  value  of  a',  as  will  make 
1  -.2jr-f  3a:-— 4j^-t-5.T''  a  square. 

Here,  the  first  term  !,  being  a  square,  let  ]  —  2a-f-3a;- 
-  ^x^-\-5x'  =  {  1  -  x-}-x-y=  1  —  2x43a=~  2a;H a^  agreeably 
to  the  method  in  Case  1. 

Then  we  shall  have  di'^-—4x^=x'^-^^x^. 

And,  consequently,  5a:--4=a:-— 2  ;   whence  3*=f=^. 

And  consequently,  l~2.T-'roa='-  4x^4-5:^=  1  —  1  -{-f-.', 
5       9.. 
-f-  — =— ;   which  is  a  square  number,  as  was  required, 

2.  It  is  required  to  find  such  a  value  of  a:,  as  will  make 
ix'^^2x^  —  x^+Sx — 2  a  square. 

Here  the   first  term  being  a  square,  let  4x'  —  2.T^  —  a^+ 

cording  to  the  method  in  Case  2. 

Then,  we  shall  have  3c-2=-— rc-l — -,  or  8.t x='2 

16       256  16 


mOPHANTINE  ANALYSIS.  219 

156 
6ll£-f25     537 


t-^tt:'  Whence,  768a; -80ir=- 612+25;  and,  consequently. 


768-80     688 
Or,  if  we  putx=-,  the  formula  in  that  case  will  become 

y 

y     y     yy 

And,  therefore,  multiplying  this  by  y'*,  which  is  a  square, 
it  will  he- -),-2!j—f-\-:^f~2>/.  Where  the  first  term 
being  now  a   square,  if  the  expression,  so   transformed,  be 

resolved  by  Case  I,  we  shall  have  2/~rq7  5   ^"^  a;=-= 

--,  as  before. 

3.  It  is  required  to  find  such  values  of  x,  as  will  make 
\^\-^x-\^'lx^ — Jx^+4.r^  a  square. 

Here,  both  the  first  and  last  terms  being  squares,  let  I  -f* 

3x--f7x2-2x^+4x«  =  (l+i:+2a-2)2=l4-3x+^a;=+6x« 

-^4x^,  according  to  the  method  in  Case  3. 

25 
Then,  we  shall  have  6a?^+  — a-^  =--  7ar-— 2x^ ;    or  6a;  -j-  2 

4 

^     25  ^    ,         ^       .  3 

%~1 — —  ;  and,  by  reduction,  «;=— . 

And,   if  we  put   the    same  formula,  1  4-3a;4-7x^ — 2x^4" 

4a;'=(  1 4- 2 J—  "ix'^)-—  \-\-.^x  —  fx"^  -  k^x^^^x^  we  shall  have, 
by  cancelling,   7a;^  — 23:-^=     |x^  — 6x^  ;  whence  b.r — 2x= 

35  35 

^I-7=--;or.=  --. 

And,  in  a  similar  manner,  other  values  of  x  may  be  found, 
by  employing  the  method  of  substitution  pointed  out  in  the 
latter  part  of  Case   i. 

4.  It  is  required  to  find  such  values  of  x  as  will  make 
2.r*—  I  a  square. 

Here,  1  being  an  obvious  value  of  .r,  let,  according  to 
Case  4,  a=l-f  ?/. 


220  DIOPHANTINE  ANALYSIS. 

Then2a;*— 1=2(1+2/)'— •=2(14-42/  +  e;2/2+4^3_j.^4._ 
l  =  l+8yH-12^^+'  r+^y^  And  .«mce  the  first  term  ot 
this  last  expression  is  now  a  square,  we  shall  have  by- 
Case  I,  l-h8i/4-i2?/2+&^ -f-2i/^=(l+4^— 22/=^)^=l-^-8l/^- 
Whence,  as  the  three  first  terras  of  tho  two  numbers  ol 
this  equation  destroy  each  other,  there  will  remain  4?/'* — • 
lby^=2y'-\-hy'  ;  or  y  12  ;  and,  c<»ii.-equently,  a==l+2/ 
=  13  ;  which  value  being  substituted  far  x,  makes  2x^  —  1 
=57121  -  (239  r,  as  required.  And  il"  1 3  be  now  taken, 
as  the  known  value  of  x,  and  the  operation  be  repeatr^d  a.s 
before,  we  shall  •>btHin,  for  another   value   of  x,  the  com- 

i0du74f,97H9 

plicated  Iraction  —      —  - . 

^  14I719J1)9 

PR  'BLEM    IV. 

To  find  such  values  of  a  as  will  make  ^(az^-[-bx^-hcx 
-^-d)  rational,  or  ox  +/>r"T-cx-|-^  ~  a  cube.  This  formula, 
like  the  two  latter  of  those  relafinii  to  squares,  cannot  be 
resolved  by  any  direct  method,  except  in  fhe  cases  where 
the  first  or  last  (eriu»  of  the  expression  are  cubes  ;  it  be- 
ing necessary,  in  all  the  rest,  that  some  simple  number 
answering  the  conditions  of  the  question,  should  be  first 
found  by  trial,  before  we  can  hope  to  obtain  others  ;  but 
when  this  can  be  done,  the  problem,  in  each  of  the  cases 
here  mentioned,  may  be  resolved  as  follows. 

RULE. 

1.  When  the  last  term  d  of  the  given  formula  is  a  cube 

c 
put  it  =e'',  and  inake  r+cx  +  bx^  +  ax  ={e  -f  —xy~ 

Then,  by  expunging  the  two  first  terms  on  each  side  oi 
the  equation,    which  are  common,  there    w ill  remain  ax^-{- 

6j;^= -x^-^ ^x-;    whence,   by  division   and    reduction^ 

27p^    t^e 

we  shall  have  27ae^x-l-276c^=c^x-f9cV,  and  consequent- 


DIOPHANTINE  ANALYSIS.  221 

iy  X — ^__^  5    which  form  fails  when  the  coeffi- 

cients  b  and  c,  or  a  and  c,  are  each  equal  0. 

2.  When  the  coefficient  a   of  the  first  term  is   a  cube^ 

put    it  =f,   and  miikepx'+bx^-{'cx  +  d^{fv+~y  = 

3J 

f\e+bx'^-^^x-^~ 
•6f'        is;7r 

Then,  by  expuiiijing  the  two  first  terms  on  each  side  of 
the  equation,  as  before,  there   will  remain  cx-\-d=^jX-\' 

'^tTT  »  whence,   by  multiplying    by  27/^,   we  shall  have  27 
pcx-\-21(ip^9b''f'i-\-b\  and  consequently  a  = 

i^TTTTi'TT^-rs;:  •   which   form  likewise   fails,  when   b  and  c, 

vf\Scf-^-~b^) 

or  b  and  d,  are  each  —0. 

3.  When  the  first  and  last  terms  are  both  cubes,  puta= 

f'    and    ^/=e^   and     make   e^^+ca;4-6.TH/V=(e+/a;)3  = 

Then,  cx  +  6.r2=3/e2a'  +  3/2cx2; 

Whence,  we  shall  have  bx — '6f'^ex=^\-\fc^ — c  ;  and,  conse- 

3/e^  -  c 
quently,  x  =  -*  — 7^;  which  formula  may  also  be  resolved 

by  either  of  the  two  first  cases. 

4.  When  neither  the  first  nor  the  last  terms  are  cubes, 
letp  be  a  value  of  x,  found  by  inspection,  or  by  trials,  and 
make  ap'^'\-bfr-\-<:p-\-d=^(f. 

Then,  by  puttin<j  x=y\-p^  we  shall  have  ap'^-\-bp^-\'Cp 
^d=a{7j'\-p)'  +  b{y-{-f>y4-c{y+p)-{-d=aif  -f  {3ap-\-b) 
y"'h  {Sap^-i-^bp  +  c^y-^-op^-j-bp"  4-cp  +  d,  or  ax^ -{-bx^ 
''\-cx-{-d=ay'^-\-(Sap-{-b)y--^(Sap~-^2bp+c)y'\-q\ 

From  which  latter  form,  the  value  of  y,  and  conse« 
^juently  that  of?:,  may  be  found,  as  in  Case  1, 


u2 


222  DIOPHANTINE  ANALYSIS 


EXAMPLES. 

1.  It  is  required  to  find  such  a  value  of  x  as  will  majki- 
x^-\-x-hl  cube. 

Here,  the  last  term  being  a  cube,  let  the  root  of  the 
cube  sought  =1+10;,  according  to  Case  1. 

Then,  by    cubiuo,  we  shall  have    14-a;-l-a^=l-i-a:+4-a;- 

4—1- 3-3  .  ^ 

And,  since  the  two  first  terms  on  each  side  of  this  equa- 
tion destroy  ear-h  other,  there  will  remaiu  2:-=r^'- +  gV^'^* 

Whence,  dividiiig  by  a-,  we  shall  have -5'^^+^—  •>  or 
x'-}-9=27;  and  consequently  x  =  27 — 9=i8;  which  num- 
ber, by  substitution,  makes  I +x-|-r^=  i -}- 1^^3^4=343 
=7"^  a  cube  number,  as  ^^as  requited. 

And  if  we  now  take  this  value  of  .t,  and  proceed  accord- 
ing !o  the  method  employed  m  Case  4,  we  shall  obtain  ■x=-- 
!3782t>        ,  .  .    , 
"^T^FTT  5  which  last  number  will  also  lead,  in  like  manner, 

to  other  new  values. 

2.  It  is  required  (o  find  such  a  value  of  x  as  will  make 
a;^4- 3x2+ 1 33  a  cube. 

Here,  the  first  term  being  a  cube,  let  its  root  =  1  +.Tj 
accordmg  to  Case  2. 

Then,  by  cubing,  we  shall  have  133-r3a^-r2:^=ri4-x)- 
=  14-3x+3xHa:^ 

And  since  the  two  last  terms  of  this  equation  destroy 
each  other,  there    will  remain  l-fSx^  133,   or  3x=l33~ 

1  =  132;  whence  x=--~=44,  and  a'^+3a:^+ 133=91125 

o 

=  (45)^,  a  cube  number,  as  was  required. 

And  if  4o  be  now  taken  as  a  known  value  of  a-,  other 
values  of  it  may  be  found,  as  in  the  last  example. 

3.  It  is  required  to  find  such  a  value  of  x,  as  will  make 
6'\-2Sr+S9x^~  125a:^  a  cube. 

Here,  let  the  root  sought  ~  2 — 5x  according  to  Case  3. 
Then,  by  cubing,   we  shall  have   8-f  SSx-f  sar^- I26u;' 
==(2-.5x)^=8-602H-150x=^-J25a:^. 
Aadj  since  the  first  and  last  terras  of  this  equation  d« 


BIOPHANTINE  ANALYSIS.  223 

slroy  each   other,  there  will  remain  28x-r89x^=— 60a:4* 

Whence,  by  dividing  by  x,  and  transposing  the  terms,  we 
shall  have  150x— 89a;=28+tiO,  or  bla;=88  ;  and  conse- 

quently  x=—. 
bl 

And  as  this  formula  can  also  be  resolved  either  by  the 
first  or  second  rase,  other  values  of  x  may  be  obtained, 
that  will  equally  answer  the  conditions  of  the  question. 

4.  It  is  required  to  find  such  a  value  of  x,  as  will  make 
o^3_3x-|-7  a  cube. 

Here,  — 1  being  a  value  of  r  that  is  readily  found,  by 
inspection,  let  r=^y  —  ',  agreeably  to  Ca-«e  4. 

Then,  bv  substitution,    we  shall  have    .x^  — 3a;-|-7~2(T/ 

And  as  \\e  last  term   of  this  expression  is  a  cube,  let  8 

+dy-  Cry'-\-2y^={2-hlyf=ii-^Sy+ly'A'^yy    accord^ 

ing  to  Case  1.     Then,  by   expunging   the  equal  terms  on 

each  side,  there  will  remain  2y^  —  Qy'^—-y^-\- — y^> 

Whence,  dividin^r  by  ?A  and  reducing  the  terms,  we 
shall  have  12^^  —  384  —  // +-^24,  or  l27i/=408,-  and,  conse- 

i()8         ,         4U8  2«l 

quently  y^j^.  and  ^^^-^j-  ^  --='-^^' 

Which  number,  by  substitution,  makes  2a;^--3x-f '7== 
SXr-^Sl)'      ^     281,  451I801H      /35HV^  .3 

-TU7— -^^^T27+--20l^:7&^3-=V'T27;^^  as  requu'^ 
ed.  And,  bv  taking  this  last  as  a  new  value  of  x,  others 
may  be  determined  by  Ihe  same  method. 


PROBLKM  V. 

Of  ihe  resolution  of  double  andiriple  equalities. 

When  a  single  formula  containing  one  or  more  unknown 
quantities,  is  to  be  transformed  into  a  perfect  power, 
such  as  a  square  or  a  cube,  this  is  called,  in  the  Diophan^ 


224  DIOPHANTINE  ANALYSIS. 

tine  Analysis,  a  simple  equality  ;  and  when  two  formtilee, 
containing  the  same  unknown  quantity,  or  quantities,  are 
to  be  each  transformed  to  some  perfect  power,  it  is  then 
called  a  double  equality,  and  so  on  ;  the  methods  of  re- 
solving which,  in  such  cases  as  admit  of  any  direct  rule, 
arc  as  follows  : 

RL'LE. 

1.  In  the  case  where  the  unknown  quantity  does  not 
exceed  the  first  degree,  a<  in  the  double  quantity, 

ax-\-b=:  □,  and  rr4-d—0, 
let  the   first  of  these    formulae  ai  -r6=^^,  and  the   second 

Then,  by  equating  the  tuo  v.ilues  of  rr,  as  found  from 
these  equations,    we  shall  have  (z  -\-(i(i — bc^au^,   or  acz^ 

And  since  the  quantity  or)  the  nght  hand  side  of  this  equa- 
tion is  now  a  square,  it  only  reuiams  to  find  such  a  value  of 
z  as  will  make,  when  the  question  is  resolvable,  ncz'^-^a(ad 
— 6c)  —  n  ;  vvhich    being  done,   according   to   the  method 

z^—b 

Doinlcd  out  in  Problem  1,  we  shall  have  .t  — . 

a 

•J.  When  the  luiknown  quanfiiy  does  not  exceed  the 
second  degree,  and  is  found  m  each  of  the  terms  of  the 
two  formulge  ;  as  in  the  double  equality 

nx'^-^bx—^J,  and  ca:^4-'^J"=n. 

Let  x~^  then,  by  substitution,  and  multiplying  each  of 
the  resulting  expressions  by  y^,  we  shall  have 

from  wjjich  last  formulae,  jhe  value  of  i/,  when  the  question 
is  possible,  and  consequently  that  of  a:,  may  be  determined 
as  in  Case  I. 

But  it^  it  were  required  to  make  the  two  general  expres- 
sions 

ax''\-bj-}-c=n  ,  and  dx~-^ex-^f=C], 
the   solution  could  only  be  obtained   in    a   few  particular 
cases,   as  the  resulting  equality  would  rise  to   the  fourth 
power. 


DIOPHANTINE  ANALYSIS.  226 

In  the  case  of  a  triple  equality,  where  it  is  required  t© 
make 

ax-{-by=\jf  c'a;H-/y?/=n,  and  eT4-/y=D 
let  the  first  of  them   ax-\-by=u''^,  the  second  cx-\-dy=^v^f 
and  the  third  ex-^fij^w'^, 

Then,  by  first  eliminating;  a?  in  each  of  these  equations, 

and  afterwards  y,  in  the   two  resulting  equations,  we  shall 

have  (a/— f?e)x;'  — (c/— ;/c)w^=(a'/      be  w^, 

or,  putting  v=uz,  and  reducing   the  terms,  the  result   will 

.                                nf-  !'e          cf — '/«      w 
give  the    simple   equality--, — —z^ — --— —  ;    where 

the  right  hand  member  being    a  square,  it  only  remains  to 

[ind  a  value  of  z   that    will    make  the  left   hand   member  a 

square  ;    which,  vvhen  possible,  may  be  done  by  Problem  1, 

Hence,    havmg  z,  we    have    as  above,   v—uz;  and   the 

irst  two  equations  will  give  x= u^,  and  y= 7-^% 

a.i  —6'  "       ad — be 

ivhere  u  may  be  any  whole  or  frartional  number  whatever. 

But  if  the  three  for'nulae,  here  prop  ):?ed,  contained  only 
)ne  variable  quan  ity,  the  simple  eqoalitv  to  which  it 
fVould  be  necessary  to  reduce  them,  W(Mj|(i  rise,  as  m  the 
ast  case,  to  the  foiirth  power :  and  i>e  equally  limited 
vith  respect  to  its  solution. 

4.  In  other  cases  o  this  kind,  all  that  ran  be  done  is  to 
ind  successively  bv  the  former  rules,  several  answers, 
vhen  one  is  know  1 ;  and,  if  neither  this  nor  any  of  the 
ibove  mentioued  nodes  of  solution  are  found  to  succeed, 
he  Problem  under  consideration  ran  only  be  determined 
»y  adopting  som-^  irtidce  of  substitution  that  vvill  fulfil  one 
>r  more  of  the  re  juired  conditions,  and  then  resolving  the 
emaining  fotvnulae.  when  they  are  possible,  by  the  me- 
hods  already  d  (ivered  for  that  purpose  ;  but  as  no  gene- 
al  precepts  oau  be  given,  for  ot)tainmg  the  solution  in  this 
vay,  the  proper  mode  of  proceedini,  in  such  cases,  must 
:hiefly  depend  upon  the  skill  and  s  igacity  of  the  learner. 

EXAMPLES. 

L  It  is  required  to  find  a  number  r,  such  that  x-fl^S 
and  x-|-192  shall  be  both  squares. 


•326  DIOPHANTINE  ANALYSIS. 

Here,  according  to  Case  1,  let  a;-{-l28=ziy2,  and  X'\-I02i 

Then,  by  eliminating  r,  and  equatingr  the  result,  wet 
shall  have  w-~]'iS-  z'^     192,  or  tfi;^'  -  64    -z^ 

And,  as  the  quantity  on  the  right  hand  side  of  the  equa- 
tion IS  now  a  square,  it  only  remains  to  niake  7v^'\-64>  a 
square. 

For  which   purpose,  put  its   root  ^w-\-n  ;  then  ^2+64 
=^y2^- 27^u+7^2,  or  2?iw-\-n^=64  ;  and  consequently  a;= 
64  — n^ 
— 2^  ;  where  takinjr  n,  which  is  arbitrary,  =2,  we  shall 

_64-4     60 
nave  w —  =— =  '-5;  and  consequently  x=w^^  128 

=  152-128=225  -  128=97,  the  answer. 

2.  It  is  required  to  find  a  number  x,  such  that  x^-^-x 
and  x^~x  shall  be  both  squares. 

Here,  according;  to  case  2,  of  the  last  Problem,  let  x=i 

-;  then  we  shall  have  to  make  -+- ,  and squares  : 

y  y    y       y'    y    ^ 

or,  by  reduction, —(H-z/)z.  □,  and  ^(l-'j/)=n. 
V"  y 

Or,    since  a  square   number,    when  divided  by  a   square 
number,  is  still  a  square    it  is  the  same  as  to  make 
1 -!-/,'=□.  and  1 — y=U  ; 

For  this  purpose,  ther.  lore,  let  i  -i-y—z'-,  or  y^z^—  1  ; 
then  !  —  //=;.'-  z^:   wuch  is  also  to  be  made  a  square. 

But  as  neither  the  first  i.nr  the  lasi  terms  of  this  formula 
are  squares,  w.'>  must,  in  order  to  surreed,  find  some  sim- 
ple number,  that  will  answer  the  condition  required  ; 
which,  it  is  evident  from  inspection,  vmII  be  the  case  when 
z~- 1 . 

Let,  therefore,  z~\-w,  a;^reeab)v  to  Problem  1,  Case 
7,  and  we  shall  have  1  — ^^  "^  -  ^^--2  — (  i —tb;)  =  I +2'a' 
—  Tsy-;  or  y==w^  ~  2w  ; 

Or,  putting  l—nw  for  the  root  of  the  former  of  these  ex- 
pressions, there  will  arise,  by  squaring,  l-{-^w—7e)^=l'^ 
2nw-\-n"w'^. 

Whence,  expunging  the  1  on  each  side,  and  dividing  by 


DIOPHANTINE  ANALYSIS.  227 

w,  we  shall   have  '2  —  w=^—2n'{-n^w;    and  consequently 
_2n-\-2   ,._!__         1        _(«^-h')' 
/i=*  -h  1 '  y      tt'-  ~  2a;     4/i  -  4h3' 

where,  in  order  to  render   the  value  of  a  positive,  n  may 
be  taken  equal  to  any  proper  fraction  whatever. 

Or,    if,    for  the  sake  of  greater  generality,  —  be  sub- 

n 

stituted  for  n,  we  shall  have 

where  m  and  n  may  now  be  taken  equal  to  any  integral 
numbers  whatever,  provided  n  be  made  greater  than  /n. 

25 

If,  for  instance,  n=2  and  m=  1,  we  shall  have  x=^  ', 

169 
and  if  n=3  and  m=2,  x=-^-- ;  and  so  on,  for  any  other 

number. 

3.  It  is  required  to  find  three  whole  numbers  in  arith- 
metical  progression,  such,  that  the  sum  of  every  two  of 
them  should  be  a  square. 

Let  X,  x-\-y,  and  a:-|--i/'  be  the  three  numbers  sought  ; 
and  put  '2a;+2/=«^  2x4-!^i/="^^  and  2.r-{-3j/— ijy^^  agreea- 
bly to  Case  '6. 

Then,  by  eliminating  x  and  y  from  each  of  these  equa- 
tions, we  shall  have  v^ — ir  —  xs)'^  —  v^,  or  2v^  —  vr=^w^. 

And,  if  we  now   put  v-~uz,  there  will  arise  ^it^z^ — u^=:. 

"jc^  \  or,  by  dividing  by  w^,  2z" — 1  =  —  ;    where,   the  right 

hand    member  beintj  a  square,  it  only  remains  to  make  2z- 
—  I  a  square,  which  it  evidently  is  when  r=  I. 

But  as  this  vahie  would  be  found  not  to  answer  the  con- 
ditions of  the  question,  let  z—- 1 — p  ;  then  i^z^—  1  =2(  1  — 
ipf—  !•=!— 4/?-|--;A 

And,  consequently,  if  this  last  expression  be  put  =(1  — 
npy^  we  shall  have,  by  i^quaring,  1  —  4p-f2/>"=  1— 2w2>4~ 
u*/>^,  or  -4-l-2/>=- — 2/1 -fn^/?  ;  whence 

2n-4        ,  2w— 4     n^— ®7i-f2 


228  DIOPHANTINE  ANALYSIS. 


Or,  if,  for  the  sake  of  greater  generality,  —  be  substituted 

for  w  in  this  last  expression,  we  shall  have 
///^-  2nin-\-2n- 

And  since,  by  the  two  first  equations,  y=v'^-^u^=uh''' 
«-w2=(2-— l)w^  and  T  =  i(M- — ^)  =  4(:^--2-^)w^,  it  is  evi- 
dent that  z  must  be  some  number  greater  than  1,  and  less 
than  ^/2. 

If,  therefore,  m  =  9  and  n=5  we  sha'l  have 
8i — i^O+oO      -^1  241      m2  720 

Sl-5()  31'         3|2       2  ^     312 

Or,  taking  v/=2X3l,  a:=482,  and  2/=2880,  we  have 
a;=482,  x+y=S'662,  and  rc-f  2^=6^:42,  which  are  the 
numbers  required. 

4.  It  is  required  to  divide  a  given  square  number  into 
two  such  parts,  that  each  of  them  shall  be  a  square.* 

Let  a^=  given  square  number,  and  x^  and  a^—  x^  its 
two  parts.  Then  since  x"  is  a  square,  it  only  remains  to 
make  a^— a:^  a  square. 

For  which  purpose  let  its  root  =«a;  —  a,  and  we  shall 
have    or — x~  =  n^x^'—'Ianx  +  «"}  or    — x^  "=■  n^x"^ — 2anx  ; 

whence,  by  reduction,  x^^  2-ui'         ^^^^  of  the  first  part. 

2«n2  an^  —  ^    ,  ^    , 

and  nx — a= — ; a=    „  ,   -  the  root  oi  the  second. 

^         „        /  2(in  \  '        ,  /on- — a\  „ 

Therefore  I  I  ^  and  I  — -—  I  ^  are  the  parte  re- 

quired ;  where  a  and  n  may  be  any  numbers  taken  at 
pleasure,  provided  n  be  greater  than  1. 

6.  It  is  required  to  divide  a  given  number,  consisting  of 

*  To  this  we  way  add  the  followintf  useful  properly  : 

If  s  and  r  be  any  two  unequal  numbers,  of  which  s  is  the  greater,  it  can 
then  be  rtadily  shown,  from  the  nature  of  the  problem,  that 

Urs,  s2—r2,  and  s2  -i-rS 
will  be  the  perpendicular,  base,  and  hypothenuse  of  a  right-angled  triangle. 

From  which  expressions,  two  -qiiare  numbpr«  may  be  found,  whose  sum 
91  difference  shall  be  square  numbers;  for  (2rs)2  -^(sa — r2)2— («3 — r2) 
2,  and  (sa -i-r2}2— (2r5)2  s=(52— r2)i,  or  (52 -}-ra)8— (53— y2)3 -s 
>2r.s)2  ;  where  s  and  »•  may  be  any  numbers  whatCT-er. 


DIOPHANTINE  ANALYSIS.  229 

two  known  square  numbers,  into  two  other  square  num- 
bers. ^ 

Let  a^-\-b^  be  the  given  numbers,  and  x^,  y^,  the  two 
required  numbers,   whose  sum,  a:"+j/"  is  to  be  equal  to  ci" 

Then  it  is  evident,  that  if  x  be  either  greater  or  less 
than  </,  y  will  be  accordingly  less  or  greater  than  b.  Let 
therefore  a' =^ a -|-"t'2',  and  u  —  b  —  nz^  and  we  shall  have  a- 

Or,  by  transposition  and  rejecting  the  terms  which  are 
common  to  each   sid(^  of  the    equation,  mV-{-M^2^=2tnz 
— tamz^  ox  m^z-\-n^z=-ibn-'2nm\.  whence 
tbn  -~2nm  . "ibmn-^a^n^ ^m^)     2amn-\-b{mP"  rr) 

where  m  and  n  may  be  any  numbers,  taken  at  pleasure, 
provided  their  assumed  values  be  such  as  will  render  the 
values  of  x,  y,  and  z,  in  the  above  expressions,  all  posi- 
tive. 

6.  It  is  required  to  find  two  square  numbers,  such  thai 
their  difference  shall  be  equal  to  a  given  number. 

Let  d^^  the  given  difference  ;  which  resolve  into  two 
factors  a,  b  ;  making  a  the  greater  and  6  the  less. 

Then,  putting  .t=  the  side  of  the  less  square,  and  x-\- 
6=  side  of  the  greater,  we  shall  have  (a;-|-6)^— x^— a^-f- 
2bx-\-b^-x-=d{ab)  or  ^Zbx-\^b^=:d{ab). 

Whence,  dividing  each  side  of  this   equation  by  b,  we 

shall  have  x~——-~  the  side  of  the  less  square  sought, 

and  x-Yb^^—- — \-b= =  the  side  of  the  greater. 

If,  for  instance,  (?— 60,  take  a  X  6=30X2,  and  we  shall 

have  x=:?5^=14,  and  24-2=^^^=16,  or    16--14'^ 

=256  —  196  =  60  the  given  difference. 

7.  As  an  instance  of  the  great  use  of  resolving  formulse 
♦f  this  kind  into  factors,  let  it  be  proposed,  in  addition  to 
what  has  been  before  said,  to  find  two  numbers  x  and  f/. 


230  DIOPHANTINE  ANALYSIS. 

such  that  the  difference  of  their  squares,  a:"— 2/^,  shall  ht'\ 

an  integral  square.  1 

Here  the  factors  of  a:- —^2,  being  a:-{-y  and   ar  —  i/,  we; 

shall  have  (a:  +  i/)X(a?  — 2/)'=a?^— 2/**     -^"^  since  this  pro-; 
duct  is   to  be  a  square,  it   will  evidently   become   so,  by 
making  each  of  its  factors  a  square,  or  the  same  multiple 
of  a  square. 

Let  there  be  taken,  therefore,  for  this  purpose, 
x-\'y=^imr,  x  —  y^ma^. 

Then,  by  the  question,  we  shall  have  {^•\-y)y^{x  —  y) 
or  its  equal  x^  —  y''=ni^r^s" ;  which  is  evidently  a  square, 
whateyer  may  be  the  values  of  n.,  r,  s. 

But,  by  addition  and  subtraction,  the  above  equation? 
give,  when  properly  reduced, 

in  {  r^-r  5^)  m  ( r^ — s^) 

where,  as  above  said,  m,  r,  and  s,  may  be  assumed  at  plea 
sure.     Thus,  if  we    take   /ai=2,  we  sliall  have  x  =  r'^-\-s-. 
and  y=r^—s^^  which  expressions   will    obviously  give  in- 
tegral values  of  X  and  y,  if  r  and  s  be  taken  =  any  integral 
numbers. 

8.  It  is  required  to  fuid  two  numbers,  such  that,  if  ei- 
ther of  them  be  added  to  the  square  of  the  other,  the 
sums  shall  be  squares. 

Let  X  and  y  be  the  numbers  sought  ;  and  consequently 
i~-hy  and  J/^-f-a;  the  expressions  that  are  to  be  transformed 
into  squares.  Then,  if  r  — .r  be  assumed  for  the  side  of  the 
first   square,  we  shall    have  x^-\-y~r-'-  *2rx-}-x-,  or  y=^r^ 

— 2rx  ;  and  consequently  ^=-^5 —  • 

And  if  5+^  be  taken  for  the  side  of  the  second  square^ 

r"  —  y 
we  shall  have  y^-r— — -=s^-\-2sy  -f  y^  ;    or,    by   reducing 

the  equation,  r^  -'y=4:rsy-i-2rs'^,  and  consequently,  by  re- 

,       .  r^'^'^rs^         ^  2rh+s' 

duciion.y—- -—,  and  x=- — — —  ;     where    r   and   -^ 

^       4rs+l  4rs-t-l 

may  be  any  numbers,  taken  at  pleasure,  provided   r  be 

greater  than  2s^, 


BIOPHANTINE  ANALYSIS.  23. 

'^*  it  is  required  to  find  two  numbers,  such  that  their 
3Um  and  difference  shall  be  both  squares. 

Let  a;  and  x^'—x  be  the  two  numbers  sought;  then, 
iince  their  sum  is  evidently  a  square,  it  only  remains  to 
Tiake  their  difference,  x'^ — 2x,  a  square. 

For  this  purpose,  therefore,  put  its  root  =x — rand  wo 
jhall  have  x^ — 2r=x^--  •rx-{-r^ ; 

Or,  by  transposition,  and  cancelling  x^  on  each  side  of 
he  equation,  2rx  —  2x=^r^ :  whence 


x=^~ -,  and 

4>r —  ' 


rvhere  r  may  be  any  number,  taken  at  pleasure,  provided 
t  be  greater  than  2. 

10.  It  is  requued  to  find  three  numbers,  such  that  not 
)nly  the  sum  of  all  three  of  them,  but  also  the  sum  of 
jvery  two  shall  be  a  square  number. 

IaqI  4x,x^ — 4x,  and  2x-\-],  be  the  three  numbers  sought  : 

}ien    4.r4-(x'-4x)=x^    (a-2-4j-)  +  (2i-{- !)==x--*2x+ J, 

ind  4x-\-{x-^ — 4r)  4-(2.r4-i)  =  '"^  +  ^x-l-  I,beinga!l  squares, 

t  only  remains  to  make  4x'  +  (2r-|-  l),  or  its   equal,  fclx-|-l, 

i  square.     For  which  purpose,  let  6x+l- n^,  and  we  shall 

n^-  J 
[lave,  by   transposition    and    division,  x= — : — ,     whence, 

6 

2 

-  +15  or  their  equals 


^      ,  36  '  """       3    '    ^^^  ^^®   numbers    re- 

quired. 

Where  n  may  be  any  number,  taken  at  pleasure,  pro- 
vided it  be  greater  than  o. 

QUESTIONS  FOR  PRACTICE. 

1 .  It  is  required  to  find  a  number  x,  such  that  x-j- 1  and 
r — 1  shall  be  both  squares.  Ans.  x=f. 

2.  It  is  required  to  find  a  number  x,  such  that  a:-|-4  and 
r-i-7  shall  be  both  squares,  Ans.  f|. 

3.  It  IS  required  to  find  a  number  x,  such  that  lO-j-a: 
and  10 — x  shall  be  both  squares.  Ans.  x=6. 


232  DIOPHANTINE  ANALYSIS. 

4.  It  is  requirt^d  to  find  ^  number  x,  such  thata--f^ 
and  x  +  l  sha!!  b<'  both  s;quHje<!.  Ans.  "/• 

5.  It  IS  reqinred  to  find  three  integral  sqvmre  numberSj 
such  that  the  sum  of  every  two  of  them  shall  be  squares. 

Ans.  528,  6796,  and  6325. 

6.  It  is  required  to  find  two  numbers  x  and  y,  such  that 
x^-\-y  and  y^-^-x  >hall  be  both  squares. 

Ans.  x~-^%,  and  y'=Y\- 

7.  It  is  required  to  find  three  integral  square  nunibers- 
ihat. shall  be  in  harmonn  al  proportion. 

Ans.  25.  49,  and  1225. 

8.  It  is  required  to  find  three  n.tejiral  cube  numbers. 
x^,  y'^,  and  z  ,  vvl!o>e  sum  niay  be  equal  to  a  cube. 

Ans.  3,  4,  and  5. 

9.  It  is  required  to  divide  a  uiven  squaie  number  (100) 
into  two  yuch  parts  thai  each  of  them  may  be  a  square 
number.  Ans.  64,  and  ^.6. 

10.  It  is  required  to  find  two  nimibrrs,  such  that  their 
difference  may  be  equal  t<j  the  difference  ot  their  squares, 
and  that  the  sum  ot'  their  squares  shall  be  a  square  num- 
ber. .  iVns.  -i  and  ^. 

11.  To  find  two  numbers,  such  that  if  each  of  them  bc- 
added  to  their  product,  the  sums  shall  be  both  squares. 

Ans.  I  and  |. 

12.  To  find  three  square  numbers  in  arithmetical  pro- 
gression. Ans.  1,  25,  and  49. 

1 3.  To  find  three  numbers  in  anthnietual  progression,, 
such  that  the  sum  of  evtry  iwo  of  them  shall  be  a  square 
.number.  'Aus.   I2et^,  84(ii,  and  1560^. 

t4.  To  find  three  nufobers  such,  that  if  to  the  square  of 
each  the  sum  of  the  other  two  be  added,  the  three  sums 
shall  be  all  squares.  Ans    >,  I,  and  '3^. 

!5.  To  find  '.wo  nun. hers  in  proportion  as  8  is  to  15, 
and  such  that  the  sum  of  their  squares  shall  be  a  square 
number.  Ans.  576  and  1080. 

16.  To  find  two  numbers  such,  that  il  the  square  of 
each  be  added  to  their  product,  the  sums  shall  be  both 
?quares.  Ans.  9  and  16. 

17.  To  find  two  whole  numbers  such,  that  the  sum  or 


DIOPHANTINE  ANALYSIS.  233 

difference  of  their  squares,  when  diminished  by  unity  shall 
be  a  square.  Ans.  8  and  9. 

18.  It  is  required  to  resolve  4225,  which  is  the  square 
of  65,  into  two  other  integral  squares. 

Ans.  2704  and  1521. 

19.  To  find  three  numbers  in  geometrical  proportion, 
such  that  each  of  them,  when  increased  by  a  given  num- 
ber (19),  shall  be  square  numbers.      Ans.  bi,  f,  and  y^f^-. 

20.  To  find  two  numbers  such,  that  if  their  product  be 
added  to  the  sum  of  their  squares,  the  resuh  shall  be  a 
square  number.  Ans.  5  and  S,  8  and  7,  1(.  and  5,  &c, 

21.  To  find  three  whole  numbers  such,  thai  if  to  the 
square  of  each  the  product  of  the  other  two  be  added,  the 
three  sums  shall  be  all  squares.  Ans.  9,  7--5,  and  328. 

t'2.  To  find  three  square  numbers  such,  that  their  sum, 
when  added  to  each  of  their  three  sides,  shall  be  all  square 
numbers. 

,   Ans.  -^VVVn*  imh  and  J-|ffi  =roots  required, 

23.  To  find  three  numbers  in  geometrical  progression 
such,  that  if  tlie  mean  be  added  to  each  of  the  extremes, 
4he  sums,  in  both  cases,  shall  bo  squares. 

Ans.  5,  20,  and  80, 

24.  To  find  two  numbers  such,  that  not  only  each  of 
them,  but  also  their  sum  and  their  diflference,  when  in- 
creased by  unity,  shall  be  all  square  numbers. 

Ans.  3024  and  5624= 

25.  To  find  three  numbers  such,  that  whether  their 
sum  be  added  to,  or  subtracted  from,  the  square  of  each  of 
them,  the  numbers  thence  arising  shall  be  all  squares. 

Ans.  V<V%  VeS  and  ^V^ 

26.  To  find  three  square  numbers  such,  that  the  sum 
«f  their  squares  shall  also  be  a  square  number. 

Ans.  9,  16,  and  ^V' 

27.  To  find  three  square  niigibers  such,  that  the  differ- 
ence of  every  two  of  them  shail  be  a  square  number. 

Ans.  485809,  342?5,  and  23409. 
29.  To  divide  any  given  cube   number  (8),  into  three 
ather  cube  numbers.  Ans,  1,  -f-f*  ^nd  ^^ . 

x2 


28i    SUMMATION  OF  INFINITE  SERIES. 

29.  To  find  three  square  numbers  such,   that  the  diffei 
ence  between   every  two  of  them  and  the   third  shall   be  a 
square  number.  Ans.  149^,  241^,  and  269'. 

30.  To  find  three  cube  number^;  -uch,  that  if  from  each 
of  them  a  given  number  (1)  be  subtracled,  the  sum  of  the 
remainders  shall  be  a  square  number. 

An5.  A |if ,  V^^,  and  b 

GF  THE 

SUMMATION  AND  INTERPOLATION  OF 
INFINITE  SERIES. 

The  doctrine    of  Infinite    Series  is  a  subje(^t  which  has 
engaged  the  attention  of  the  greatest  mathemnticians,  both    ^ 
of    ancient   and    niodern    times  ;    and    when  taken  in  it*  ij 
whole  extent,  is,   perhaps,   one    of  the  most   abstruse  and 
difficult  branches  of  abstract  mathematics. 

To  find  the  sum  of  a  series,  the  number  of  the  terms  of 
which  is  inexhaustible,  or  infinite,  has  been  regarded  by 
some  as  a  paradox,  or  a  thing  impossible  to  be  done  ;  bui 
this  difficulty  will  be  easily  renjoved.  by  considering  that 
every  finite  magnitude  whatever  is  divii^ible  in  infinitum, 
or  consists  of  an  indefinite  number  of  parts,  the  aggregate, 
or  sum  of  which,  is  equal  to  the  quantity  first  proposed. 

A  number  actually  infinite,  is,  indeed,  a  plain  contradic- 
tion to  all  our  ideas  ;  for  any  number  that  we  can  possibly 
conceive,  or  of  which  we  have  any  notion,  must  always 
be  determinate  and  finite  ;  so  that  a  greater  may  still  be 
assigned,  and  a  greater  afier  this  ;  and  so  on,  without ^a 
possibility  of  ever  coming  to  an  end  of  the  increase  or  ad- 
dition. 

This  inexhaustibility,  therefore,  in  the  nature  of  num- 
bers, is  all  that  we  can  distioctly  comprehend  by  their  in- 
finity :  for  though  we  can  easily  conceive  that  a  finite 
quantity  n.ay  become  gre-Uer  and  greater  without  end,  yci 
we  are  not,  by  that  means,  enabled  to  form  any  notion  of 
the  ultimatum^  or  last  magnitude,  which  is  rncapable  t'f 
fartlier  aucrmenlation. 


SUMMATION  OF  INFINITE  SERIES.    ^5 

Hence,  we  cannot  apply  to  an  infinite  series  the  common 
notion  of  a  sum,  or  of  a  collection  of  several  particu- 
lar numbers,  which  are  joined  and  added  together,  one  af- 
ter another  ;  as  this  supposes  that  each  of  the  numbers 
composing  that  sum,  is  knoyvn  and  Hetermmed.  But  as 
every  series  gent^rally  observes  some  regular  law,  and  con- 
tinually approa<hes  towards  a  term,  or  limit,  we  can  ea- 
sily conceive  it  to  be  a  whole  of  its  own  kmd,  and  that  it 
must  have  a  certain  real  value  whether  that  value  be  de- 
terminable or  ijot. 

Thus  in  rnuny  series,  a  number  is  assignable,  beyond 
which  no  nuinl>er  of  its  terms  can  ever  reach,  or  indeed, 
be  ever  pertt-ctly  equal  to  it ;  but  yet  may  approach  to- 
wards it  in  su'-.h  ,i  maimer,  as  to  differ  fro.K  it  by  less  than 
any  quantity  ihat  can  be  named.  So  that  we  may  justly 
call  this  the  value  or  sum  of  the  series  ;  n(jt  as  being  a 
number  found  by  the  common  method  of  a^idition,  but  such 
a  limitation  of  the  value  of  the  series,  taken  in  all  its  infi-^ 
nite  capacity,  that,  if  it  were  possible  to  add  all  the  terms 
together,  one  after  another,  the  sum  would  bo  equal  t« 
that  number. 

In  other  series,  on  the  contrary,  the  aggregate,  or  value 
©f  the  several  terms,  taken  collectively,  has  no  limitation  ; 
which  state  of  it  may  be  expressed  by  saying,  that  the  sum 
«f  the  series  is  intinitely  great;  or,  that  it  has  no  deter^ 
minate  or  assignable  value,  but  may  be  carried  on  to  sucb 
a  length,  thai  its  sum  shall  exceed  any  given  number  what- 
t^ver. 

Thus,  as  an  illustration  of  the  first  of  these  cases,  it 
may  be  observed,  that  if  r  be  the  ratio,  g  the  greatest 
term,  and  /  the  least,  of  any  decreasing  geometric  series, 
the  sum,  according  to  the  common  rule,  will  be  {rg — /)-i- 
(r — 1)  :  and  if  we  suppose  the  less  extreme  /,  to  be  dimi- 
nished till  it  becomes  =0,  the  sum  of  the  whole  series  will 
lie  rg-^[r^  1) :  for  it  is  demonstrable  that  the  sum  of  no 
assignable  number  of  terms  of  the  series  can  ever  be  equfil 
to  that  quotient ;  and  yet  no  number  less  than  it  will  ever 
fib  equal  to  the  value  of  the  series. 

Whatever  gonsequcnc^s,  thereforf ,  follow  from  t*he  sKJv 


'^36     SUMMATION  OF  INFINITE  SERIES. 

position  of  Tg-^ij — 1)  being  the  true  and  adequate  value 
of  the  series  taken  in  all  its  infinite  ca  acity,  as  if  all  the 
parts  were  actually  determined,  and  added  together,  no  as- 
signable error  can  possihly  arise  from  them,  in  any  ope- 
ration or  demonstration  where  the  sum  is  used  in  that  sense  ; 
because,  if  it  should  be  said  that  the  series  exceeds  that 
value,  it  can  be  proved,  that  this  excess  must  be  less  than 
any  assignahle  difference  ;  which  is,  in  effect  no  difler- 
ence  at  all  ;  whence  the  suppo-ed  error  cannot  exist,  and 
consequently  ri^-r('^ — ')  Jnay  he  looked  upon  as  expiess- 
ing  the  true  value  ot  the  series,  continued  to  infinity. 

We  are,  also,  farther  satisfied  of  the  reasonableness  of 
this  doctrine,  by  finding,  in  fact,  that  a  finite  quantity  is 
frequently  convertible  into  an  infinite  seiies,  as  app  ars 
in  the  case  of  circulating  decimals.  Thus  two  thiids  ex- 
pressed decimally  is  I  =.6bt6-,  &c,  =-{-^-\-  rto  +  rAo 
-|-  Yol^o  "^  ^^-  continued  ad  infiniivm.  But  this  is  u 
geometric  series,  the  first  term  of  which  is  y\,  and  the 
ratio  -Y^  ;  and  therefore  the  sum  of  all  its  terms,  conti- 
nued to  infinity,  will  evidently  be  equal  to  |,  or  the  num- 
feer  from  which  it  was  originally  derived.  And  the  same 
may  be  shown  of  many  other  series,  and  of  all  circulating 
ilecimals  in  general. 

With  respect  to  the  processes  by  which  the  sufomatioi; 
of  various  kinds  of  infinite  series  are  usually  obtained, 
one  of  the  principal  is  by  the  method  of  differences  point- 
ed out  and  illustrated  in  Prob.  iv.  next  foUowmg. 

Another  method  is  that  first  employed  by  James  and 
tiohn  Bernoulli,  which  consists  in  resolving  the  given  se- 
ries into  several  others  of  which  the  summation  is  known  ; 
ar  by  subtracting  from  an  assumed  series,  when  put  =s, 
-the  same  series,  deprived  of  some  of  its  first  terms ;  in 
which  case  u  new  series  will  arise,  the  sum  of  which  will 
be  known. 

A  third  method,  which  is  that  of  Demoivre,  consists  in 
putting  the  sura  of  the  series  =s,  and  multiplying  eaeh 
side  of  the  equation  by  some  binomial  or  trinomial  expres- 
sion, which  involves  the  powers  of  the  unknown  quantity 
j\  and  certain  known  coefficients :    then  taking  x,  after 


SUMMATION  OF  INFINITE  SERIES.     237 

the  process  is  performed,  of  such  a  vahie  that  the  assum=. 
ed  binomial,  &c.  shall  become  =^0,  and  transposing  some 
of  the  first  terms,  a  series  will  arise,  the  sum  of  which  will 
be  known  as  before. 

Each  of  which  methods,  modified  so  as  to  render  it 
more  commodious  in  practice,  together  with  several  other 
artifices  for  the  same  purpose,  will  be  found  sufficiently 
elucidated  m  the  miscellaneous  questions  succeeding  the 
following  problems. 

PROBLEM  I. 

Any  series  being  given  to  find  its  several  orders  of  dif- 
ferences. 

RULE. 

1.  Take  the  first  term  from  the  second,  the  second  from 
the  third,  the  third  from  tht-  fourth,  &c.  and  the  remainders 
will  form  a  new  series,  calU^d  the  first  orier  of  differences. 

2.  Take  the  first  terra  of  this  last  series  from  the  se- 
cond, the  second  fr«.m  the  tbird,  the  third  from  the  tburth, 
&c.  and  the  remamders  will  form  another  new  series,  call- 
ed the  second  order  of  dfferencei>. 

3.  Proceed,  in  the  same  manner,  for  the  third,  fourth} 
fifth,  &c.  order  of  diffVreoces  ;  and  so  on  till  they  termi- 
nate, or  are  carried  as  far  as  may  be  thought  necessary.* 

EXAMPLES. 

1.  Required  the  several  orders  of  differences  of  the  se- 
ries 1,  22,  32,  42,  52,  6%  &c. 

1,4,9,  16,  25,  36,  &r. 

3,5,    7,    9,  11,  &c.  1st  diff. 

2,     2,     :%     2,  &c.  2d  diff. 

0,    0,     0,  &c.  3d  diff. 

2.  Required  the  different  orders  of  differences  of  these- 
s', 4^  5%  6^  &c. 


*  When  the  several  termsof  the  series  continually  increase,  (he  ditfereqces 
\v\\\  all  be  positive  ;  but  when  thoy  decrease,  thedit?'ercnccs  will  be  negative 
»nd  jwsitive  alternatcij'. 


r  ^ 


238     SUMMATION  OF  INFINITE  SERIES- 

1,  8,  27,  64,   lf5,  216,  &c. 

7,   19,  37,     61,     91.  &c.  1st  diff. 
U,   iS,     24,     3u,  ^c.  2d  diff. 
6,       6,       6,  ^c.  iid  diff. 
0,       0,  &c.  4th  diff. 
<J.  Required  the  several  orders  of  differences  of  the  se- 
ries 1,  3,  6,  lU,  15,  2  1,  ^c. 

Ans.  1st,  2,  3,  4,  5.  &c  ;   '^d,  1,  1,  1,  &C. 
4.  Required  the  several  orders  of  differences  of  the  se- 
ries J,  6,  2'(,  50,  105,  i!?6,  &c. 

Ans.    1st,  5,  14,  .30,  55,  91,  &c.  ;  2d,  9,  16,  25, 
3  ,  &c.  ;  -id,  7,  9,  II,  &c.  ;  4rh,  2,  ^4,  &c. 
6.  Required   the  several   orders  of   differences  of  the 
.      1    1    »      f       1     . 
'^^^^^^2' 4' 8' 16'  T^^^^- 

Ans.   lst,^^,-^,-i,~&c.  ;  2d.  1,  i,  1,  &c. 

3d,--:,---,&c.;4th,— .,  — ,&c. 
16      32  lo   32 

PROBLKM   li. 

Any  series  a.  b,  c,  (J,  e^  &c.  being  given,  to  find  the  first 
term  of  the  nth  order  of  differences. 

RULE. 

Let  (5  stand  for  the  first  term  of  the  nth  differences. 

rriu  n  t    I        ^  —  ^  n-ln-.2.,         W  — 1 

Then  will  a  —  nb -{- n,——-c — n,—^ — .— ; — d-^- n. — — 

. . e,  &c.  to  n  +  l  terms  =(5,  when  n  is  an   even 

ri        4 

number. 

.     ,  ,     -  n — 1     ,      n  — Iw— 2,         n-ln-2 

And  — a-\-nb  -^rK——-C'\-n.—- — .—-__ — rf— n. — ^ — .  ^ 

^  2  3  '^3 

„    e,  &c.  to  n-\-\  rerms  =5,  when  7i  is  an  odd  number.'' 


*  When  the  terms  of  the  several  orders  of  differences  happeii  to  be  veiT 
great  it  will  be  more  convenient  to  take  the  logarithms  of  tht;  quantities  con^ 
cerned  whose  differences  will  be  •smaller:  and  when  the  operation  is  fitiistied, 
i.h§  quantity  answering  to  the  last  log-arithm  may  be  easily  foundi 


SUMMATION  OF  INFINITE  SERIES.    2S0 

EXAMPLES. 

1 .  Required  the  first  term  of  the  third  order  of  diflTer- 
onces  of  the  series,  1,  5,  15,  35,  70,  &c. 

Here  a,  6,  c,  d^  e,  &c.=  l,  5,  16,  :^5,  70,  &:c.  andn=3. 

ixru  II  n—  1  n  —  1     n — 2    ,  , 

W hence -^a-f- no —  n,-— — c-f-n.  .  — —  rf=  —  a-f- 

'3b — 3c-frf=— 14-15— 45+35=4=  the  first  term  requir- 
ed. 

2.  Required  tho  first  term  of  the  fourth  order  of  difTer- 
ences  of  the  series  1,  f^,  lil,  64,  lib,  &c. 

Here  a,  b,  c,  d,  e,  &c.  =1,8,  '27,  H4, 1 25,  &c.  and  n=4. 

„.,  ,  ,      w-1  7i~l     n—2,^      n~\ 

Whence  a  — no -{-n. — -— c  — n.  . d-f-n. 

2  2  3  2 

'^TL-  .  !^_e=a-46+6c-4fi+c=  1  ~32-f- 1 62 -256-f  125 
3  4 

—  0  ;  so  that  the  first  term  of  the  fourth  order  is  0. 

3.  Required  the  first  term  of  the  eighth  order  of  difTer- 
enaes  of  the  series,  J,  3,  9,  29,  81,  &c.*         Ans.  256. 

4.  Required  the  first  term    of  the   fifth  order  of  diflfer- 

ences  of  the  series,  1,  -,  -,  -,  -g,  — ,  — ,  &c. 

Ans.     — rrr. 

36 

PROBLEM  III. 

To  find  the  nth  term  of  the  series,  a,  b,  c,  d,  e,  &c, 
when  the  differences  of  any  order  become  at  last  equal  to 
each  other. 

IIULE. 

Let  d',  d",  ti",  </'"',  &c.  be  the  first  of  the  several  orders 
of  differences,  found  as  in  the  last  problem. 

n-l  .,  ,  n~  1  n  —  2n—l  n—2  n-3 
Ihen    will   a-f-y-riH p.— ~-f^  H r''~2~'~T 

„„  ,  71 — 1  n—2  n  —  3  n--4     ,.    ^  ,  .     ^ 

d  H '— .j-»— -r~>  ^*'''  ^^*  =wth   term   required. 

I  ^  o  4 

*  The  labour  in  questions  of  ihis  kind  may  be  often  abridged,  by  putJing 
-ipheisfor  some  of  the  tercns  at  the  beginning  of  the  series;  by  which  means 
several  of  the  differences  will  be  eqr.al  to  0,  and  theansr/eron  that  acconnf, 
obtained  in  fewer  term? 


240    SUMMATION  OF  INFINITE  SERIES. 

examplfs. 

1.  It  is  required  to  find  the  twelfth   term  of  the   series 
^,  6,  12,  20,  30,  &c. 

2,     6,      »2,     20,     30,     &c. 

4,       6,       8,      10,     &c. 

2,       2,       2,     &c. 

0,       0,     &c. 

Here  4  and  2  are  the  first  terms  of  their  differences. 

Let,  therefore,  4:  —  d\  2  =  c/",  and  n=l2. 

Then  a+— -6/'+'-^.^^"=  2+11  d'+56d"-2+ 
1  I.        iZ 

44+1 10=  l5eJ  =  15th  term,  or  the  answer  required. 

2.  Required  the  twentieth  term  of  the  series,  1,  3,  6, 
10,  15,  21,  &c. 

1,     3,     6,     10,     15,     21,     &c. 

2,     3,       4,        5,       6,     &c. 

1,        1,        1,        1,     &c. 

0,       0,       0,     &c. 

Here  2  and  1  are  the  first  terms  of  the  differences. 

Let,  therefore,  2= d',  \=d'\  and  «=20. 

Thena+^(i'+*i=^-.!i^d"=l  +  19d'+l71d"=l  + 
112  ^ 

:i8+171=2l0=20th  term  required. 

3.  Required  the  fifteenth  term  of  the  series,  1,  4,  9,  16, 
25,  36,  &c.  Ans.  225. 

4.  Required  the  twentieth  term  of  the  series,  1,  8,  27, 
64,  125,  &c.  Ans.  8000. 

5.  Required  the  thirtieth  term    of  the   series,   1,  -,  -, 

o    o 

PROBLEM  IV.* 

To  find   the  sum  of  n  terms  of  the   series,  a,  6,  c,  c/,  e. 

*  When  the  differences  in  this  or  the  former  rule  are  finally  t=  0,  any  term, 
Ok'  the  sum  of  any  number  of  the  terms,  may  be  accurately  determined;  but 
if  the  differences  do  not  vanish,  the  result  is  only  an  approximation;  which, 
iaowever,  may  be  often  very  usefully  applied  in  resolving  various  questions 
that  may  occur  in  this  branch  of  the  subject,  and  which  will  become  conti- 
nually nearer  the  truth  as  the  differences  diminish. 


SUMMATION  OF  INFINITE  SERIES.     241 

&c.  when  the  differences  of  any   order  become   at  last 
equal  to  each  other, 

RULE. 

Let  d\  d",  d"\  r/i^,  &c.  be   the   first  of  the  several   or» 
-4ers  of  differences, 

Ihen  will  na-{-n,—^ — d  -j-n.— — — . — ^— a  +n.—- — 

cr'-f-n.~— -. . ^-d'%  &c.  =  to  the 


3        4  '234 

sum  of  n  terms  of  the  series. 

EXAMPLES. 

1.  Required   the  sum  of  n  terms  of  the   series,  1,  2,  3, 
1,  5,  6,  &c. 

Here     1,     2,     3,     4,     5,     6,     &c. 
1,     1,     1,     1,     1,     &c. 
0,     0,     0,     0,     &c. 
Where  1  and  0  are  the  first  terms  of  the  differences. 
Let,  therefore,  a=l,  d'-^l,  and  d"=0. 

Ihen  will  na+n.  — ^— «  =n-\ — =  — ^ — =  sum  of  u 

terms,  as  required. 

2.  Required  the    sum   of  n  terms  of  the  series,   1^,  2^, 
.^2,  4^  5-,  &c.,  or  1,  4,  9,  16,  25,  &c. 

Here  1,     4,     9,     16,     25,     &;c. 
3,     6,       7,       9,     &c. 
2,       2,       2,     &c. 
0,       0,     &c. 
Where  3  and  2  are  the  first  terms  of  the  differences. 
Let  therefore,  a=],  d'=3,  and  rf"=2. 
?i  —  I   ,  ,      ?i  —  1  w  —  2  , 
Then  will  na+n  .  — ^—^  +^'— q~--o~^'  ~  '^^  +3n. 

,i__l            n-l  ti-~2     3rt2— 371  ,  n3-37i2_f-2n 
-^+'"-  -  2— T-=^— ^ 3 == 

•iX(n-f  l)X(2M-f  1)  „     ^  .     , 

— i ^-i ^=  sum  of  n  terms  as  required. 

•6 

Y 


242     SUMMATION  OF  INFINITE  SERIES. 

3.  Required   the    sum  of  u   terms   of  the  series    1^  2  . 
3^,  4^  5^  &c.  or  1,  8,  27,  64,  I'id,  &c. 

Here  1,     8,     27,     64,     125,  ^c. 

7,     19,     37,       61,  ^c. 

12,     18,       24,  Src 

6,  6,  ^-c. 

0,  4-c. 

Where  the  first  terms  of  the  differences  are  7,  12,  and 

6. 

Let,  therefore,  a  =  l,  d'  =  7,  (^"=12,  and  c/"'=6. 

Then  will  na-\-n. — - — 1/4-^--7 — • — ^ — "  4""* — : — 

/i  — 2     n-3„„         ,„     n-1,    ,^     7^ — 1     n — 2      .      ^ 
__.  .  __,"=„+7„._+  12«.-—  .  —    +    6n  ^ 

?i  — 1  «  — 2  n — 3  ,  In^ —  In 

.___.__.-__=„ H f.  2n3  —  6n2  +  4n  + 

n*_6n^+llrr  — 6n_4n     147r-l4ri     Sti^- 24n-+167i 

1  l""^  1  ^  4  ^ 

n^-6n=^+lJn^-6n      n^+2n^+n2 

^ = =    sum   of  n  terms   as 

4  4 

required. 

4.  Required  the  sum  of  n  terms  of  the  series,  2,  6,   12, 

'20,30,4-0.  Ans.  !2l'i±il^+i\ 

5.  Required  the  sum   of  n  terms  of  the  series,    1,  3,  (i, 
10,  la,  4-c.  Ans.  r.-^— .— TT-. 

6.  Required  the  sum  of  ri  terms  of  the  series,  1,  4,  10, 

20,35,4-0.  Ans.!-".4-i.?±^.!i±^. 

\      Z         3         4 

7.  Required  the   sum  of  n  terms    of  the  scries    V,  2', 
3^4^  t^c,  or  1,  16,  81,  256,  <S^c. 

Ans.  —  + — 4- . 

6       2       3      30 

6.  Required  the  sum  of /^  terms  of  the  series  P,  2',  3": 
4,0,4...  Ans.-+-+y2--^ 


SUMMATION  OF  INFINITE  SERIES.     24ii 


PR>>»BLEM   V. 


The  series  a,  b,  c,  li,  e.  &c.  being  given,  whose  terms 
are  an  unit's  distatjce  from  each  other,  to  find  any  inter- 
mediate term  by  interpolation. 


RULE. 


Let  ,r  be  the  distance  of  any  term  y,  that  is  to  be  inter- 
polated, frofu  the  fir-t  term,  and  d',  d'\  d'",  &c.  the  first 
terms  of  the  dirierences. 

X 1  X 1  X — 2  x^l 

Then  will  a-{-xd-\-x.~--—d"-rx.—^^.^—^d"'-\-x.—-- 


x—2  x—3  ,      „ 
^     .  d    ,&c. 


EXAMPLES. 


1.  Given  the  loijarithmic  smes  of  V  0',  1<=  T,  V  2',  and 
r  3'.  to  find  the  lo^r.  .sine  of  1"  1'  40". 

Here      r  0'  1     .'  V  2'  1^  3' 

Sines  8.241855.3       8.24903:^2      8.256' »943       8.2630424 

71779  70611  69481 

—  ;i68  —1130 

38 

Whence  the  first  terms  of  the  differences  are  71779,— 

11 6 J^,  and  38. 

Let,  therefore,  x  =  l'^'  1'  40" --l^  0'=1'  40"=1|  =  dis- 
tance of  y,  the  term  t(.  be  interpolated  ;  and  fi'=71779,  d" 
=—1168,  and  d'"^^8. 

J. 2  X  —  1  X'  —  2 

Then  will  y  =a'{-xd' -\-x.'——  d" -\'X.'—--.——d."' =a-\- 

2i  ^  o 

^d'+^d" ^—d'" -^8.24  \S553  +  .0I196S1  +  0000694  - 
3        9         SI  ^ 

.0000002=8.'e53R876=sineof  1°  1' 40",  as  was  required. 

^,  Given  the  series  ^,,it»^i;>^»7t>  &c.  to  find  the  term 
50  51  52  63  54 


244     SUMMATION  OF  INFINITE  SERIES. 

which  stands  in  the  middle  between   the  two  terms  --  and 

52 

1_  J._ 

53*  105' 

3.   Given   the  natural  tangents  of  88^  h4',  88<>  55',  88*^ 

56 ,  88    57',  88°  58',  88°  59\  to  find  the   tangent  of  88^ 

58  11".  Ans.  55.711144. 

PROBLEM  IV. 

i 
Havinor  given  a  series  of  equidistant  terms,  a,  b,  c,  d,  e,' 

(fee.  whose  first  difierenf-es  are  small  ;  to  find  any  interme- 
diate term  by  interpolation. 

RULE, 

Find  the  values  of  the  unknown  quantity  in  the  equation 
which  stands  .laainst  the  given  number  of  terms,  in  the  fol 
lowing  table,  and  it  will  give  the  term  required.* 

1.  0—6X40 

2.  o_.^6+c=0 

3.  a—3h-\-Sc—d=0 

4.  „ — ib-\-6c-4d+e=0 

5.  «— 56-f  10c— 10rf-|-5e— /=0 

6.  o—>^b-rl5c-20d-\-l5e  -  e/A-g-O. 
I  n— I  n—l     n — 2 

Or    I  2  ^^  3      ^ 

".-^.-^-.-^e,&G.  =0. 

EXAMPf  ES. 

1.  Given  the   logarithms  of  101,  102,  1  04,and  105,  to 
find  the  logarithm  of  103. 


*  The  more  terms  are  given,  in  any  series  of  this  kind,  the  more  accu- 
rately will  the  equation  that  is  to  be  used  approximate  towards  the  true  v 
suit,  or  answer  required, 


SUi\IiVI\T[ONr  OF  IVFINITE  SERIES.     245 

Hsre  the  nuinj  •<  of  terms  is  4. 
And  against  4,  in  the  tubl^',  we  have  a  — 46-|-'ic  — 4c/-|-e 

=0  ;  or  c= ■ —   - — '-Uz=:  value  of  the  unknown 

H 

quantity,  or  term  to  be  to'irid. 

a=2.00432l4 
/■=2.0086  '02 
^/=2.0!7<)333 
6  =  2.0211893 
And  consequently 
4X{b-\-<l)  =lo.  1025340 
a-f-e  ~  4.0255iU7 


Where,  takin^  the  lo^.-s.  of 
101,  102,  104,  and  105 


0)1^0770233 

2.or28372=log.  of  103,  as 
required. 

2.  Given  the   cube  roots  of  45,  46,  47,  48,  and  49,  to 
find  the  cube  root  of  50.  Ans.  3.684031 . 

3.  Given  the   lo;^arit;ims  of  50,  51,  5'2,  54,  55,  and  56, 
to  find  the  logarithm  of  53.  Ans.   1.7242758695. 

PROMISCUOUS  EXAMPLES  RELATING  TO  SERIES. 

1.  To  find  the  sum  fs)  of  n  terms  of  the  series,  1,  2,  3. 
4,  5,  &c. 

First,  1  +  2+34-4+5  &c.      .     .      .     n=s. 
And  /i+(n— l)+(n-2)  +  (n— 3)+(n-4)  &c. 

+l=s; 

Therefore,  by  addition, 

(n  +  l)  +  (/i  +  l)  +  (ri  +  l)  +  (/i+l)  +  (n  +  l)  &C 

+(n  +  l)=r2s. 

And   consequently   ?i(w+l)==2s  ;  or  s= — - —  =  sum 

required. 

2.  To  find  the  sum  (s)  of  n  terms  of  the  series,  I,  3,  5, 
7,  9,  11,  &c. 

First,  1+3  +  5+7  +  9  &c (2n— l)=^s. 

And  (2/t— l)  +  (2rt-3)+(2rt~5)+  .  .  .   +I  =  s. 
v2 


\!46     SUMMATION  OF  INFINITE  SERIES. 

Therefore  by  addition, 

2n+2n+2n-\-2n-^2n-\-  6lc 2n=2s, 

And  consequently  k;/tXn  =  2s  : 

2n2 
Or  s=.—.-=n'^=z  sum  required. 

5.  Required  the  sum  (s)  of  n  terms  of  the  series,  a-\- 
{a-\-fi)-\-lai-2J)-^{a-^3</)-{-{a-\-4<i)  &c. 

First,    a  i-  {a-\-d)-{-{a-]-2d)-{-{a  \-:iH)  &c -f- 

And  a4-  {nd^d)  -{-  a -]-  {nd-2ri)-\'a-\-{7id^3d)-{-a-{- 
(nd — 4^)  &:c a=s. 

Therefore,    by    addition,  "^a-^-^nd— d)-\-'i:a-{-{nd — d)-\- 

2a-\-(nd — li)  6ic +2a-\-{nd—d)  =  2s. 

And  consequently  (2a-\-nd~  d)  Xn=2s  ; 

n 
Or  s^={2a-\'nd — d)  X-=  sum  required. 

Or  the  same  may  be  done  in  a  different  manner,  as  fol- 
lows : 

a-\-{a-\-d)-\-{a-\-2d)-\-[a-\-^d)^{a-\-Ad)  ike. 
_  I  (+l-f  14-1-f  1  +  1  &c.)Xa 
~  I  (4-0  +  1+2  +  3-1-4  &c.)Xrt 
But  n  terms  of  1  +  1  +  1  +  1  +  1  &c.  — «. 

And  n  terms  of  0+1+2+3+4  &c.  ==!i21i^'I^'^. 

Whence  s=ria+---^"=ll^^  =  {2a  +  rf  (n  _  1  )  j  X^ 

which  is  the  same  answer  as  before. 

4.  To  find  the  sum  (s)  of  n  terms  of  the  series  1,  x,  x^. 
x^f  a:*,  &€. 

First,  1  +x'  +.T2+a;3+x\  &c a;"^'  =s. 

And     .T+j2+j:^+a:'+a;%  &c a^"=?x. 

Whence,  by  subtraction,  x^—  l=sx  — s. 

rjQn 2 

Or  s= — =  sum  required. 

And,  when  a;  is  a  proper  fraction,  the  sum  of  the  se- 
ries, continued  ad  infinitum,  may  be  found  in  the  same 
manner. 

Thus,  putting  l+a^+x^+x^-f-a^^+a;^,  &c.  =s. 


SUMMATION  OF  INFINITE  SERIES.     247 

We  shall  have  x-\-x-^  }-r^+x''-h.T%  &;c.  =sx, 
And  consequently  ~l=sr  — s;  ors-.sr  =  l, 

Whence  s= =  sum  of  an  infinite  number  of  terms, 

1  —X 

as  was  to  be  found. 

6.  Required    tne   sum    (s)    of  the    cu'culating   decimal 
,099^99  &c.  continued  a^l  infinitum. 

Firs,,  .99999.  &c.  =  1+^^  J-+ ^^6...  = 


_l+_L+JL+_L.+&o.=i. 
lU^U'O      1000      100"0^  9 


Therefore,  l  +  l^-^-+^-+&c.  =.i^. 

.     ,      ^^^s      s     9s 
iind  consequently  1  =— -=— -rrs  ; 

Whence  s=l  =  sum  of  the  series. 
G.  Required  the  sum    (s  ;  of  the   series   a^-\-{a-\-dY-{- 
a+2(i)'+(a+3^)2-f  (aH-4'/)2,  &c.  continued  to  n  terms, 
Here 
First,  a'=a" 

(rt-f4c/)2^a^+2X  m.i-M6f/- 
&c.  &c. 

Whence 
Sum  of  71  terms  of  (14-l-|-l  +  l  +  &c.)a=^ 
+  .  .  .    ditto  of  (t)-hl+2-{-3+4-l-&c.)2(£ 
+  .  .  .   ditto  of  (0-(-l-f-4+9-f  l6  +  &c.)a 
But  n  terms  of  i-f  1-j-l-f  l4-&c.=w. 

And  of  0  +  1+2+3-1-4  &c.=^'-^i^. 
Alsoof  0+H-4+9+&C.  ="i^i^^-5 


I 


248     SUMMATION  OF  INFINITE  SERIES. 


Therefore  s=na2-}-n(n- I  )acZ+"^^^^^^—^  d^    ^ 

the  whole  sum  of  the  series  to  n  terms. 

7.  Required  the  sum  (s)  of  the  series  a^+{a-\-d)^-{'{a 
+2'/.H(«+3'/)"4-((i-f-4(/)'  ^c»  continued  to  71  terms. 

First,  a'—a-^ 

(a 4-'0'='*'+-^-^ '«"'•'+■■' X  la^'+l^/' 
{a-\-2iy  =  a'-i-    X:iu2.7-f8X4aYi^+8df3 

lo-^Sif=^a'-\-^X  m-J-hSX9nd'-\-27d'' 
{a+'idy=a^-hSX  4a~d-^3  XlSad'-^-GW 
(a  -f-5d)3=a'^+3  X^a^tZ+S  X2oad^-{-  l25d  '■ 
^c.  (S,'c. 

Whence 
Sum  of  «  terms  of  (1  +  14- 1  +  1  &c.)a^ 

+  .  .  .    ditto  of  (0-{- 1+2-1-3 4-4  &c.)3a=a 
+  .  .  .    ditto  of  (0+1  +  4+9  +  16  kc.)'Sad' 
+  .  .  .    ditto  of  (0  +  1+8+27  +  64  &c.)r/- 
But  n  terms  of  1  +  1  +1  +  1  +1  &c.  =n. 

Ditto  .  .  .  of  0  +  1+2+3  +  4  &c.='^i^ 
Ditto  .  .  .  of  0+1+4+9  +  16  &c.='l^~lil^^i^ 

Ditto  .  .  .  of  0  +  1+8+27  +  64  6ic,  =  ~----^- 

72(n-l)3aV     7.(n-l)(2n-l)3aci^ 
Therefore,  s  — ;ja^H — ^ -^ -{-— ^^- + 

=sum  ol  H  terms,  as  was  to  be  found. 

8.  Required  the  sum  (s)  of  n  terms  of  the  series  1  +  3 
+7+15+31,  &c. 

The  terms  of  this  series  are  evidently  equal  to  J,  (1  + 
2),(l+2+4),  (1  +  2+4+8),  &c.  or  to  the  successive 
sums  of  the  geometrical  series,  1,  2,  4,  8,  16,  d:c. 

Let,  therefore,  a  =  l  and  r=2,  and  we  shall  have 

a-{-ar-{'ar''{-ar'-'rar^  &c.  =  1+2  +  4  +  8  + 16,  &c. 

But  the  successive  sums  of  1,  2,  3,  4,  &;c.  terms  of  this 
series  are. 


SUMMATION  OF  INFINITE  SERIES.     249 


1 =    r-.l)X-   — 

ar- — a      ^  „  a 

2.  -— -^-=(r=^-.l)X    _— 

^    «r^  -a       ^    ,                  « 
3. =  (r'_l)X     

ar* — :(  ^  a 

&c.  &,c. 

..,,        ^  «     ^   In  terms  of   r-f-r-+r^+r''  &c. 

r I      |— «Term>ot  l  +  l  +  l -f- 1  oIC.  . 

But  1  -}- 1  + 1  4-  •  +  I  -fl  H-  1  &c.  =n 

And  r+r^-^r'-^-r-^-  &o.  =  (r  —1)  X-^ 

r  —  1 

»»r,  rC'/* '  —  I )         a  a         ^, 

Whence  s=  -^ X n  X -~2{t'' —  1)  -n  - 

r  —  I  r  —  I  r —  1  ^ 

whole  sum  required. 

9.  It  is  required  to  find    he  sum  of  n  terms  of  the  series 
1,37,11,31      63    ^ 

Here  the  terms  of  this  series  are  the  successive  sums  of 

tlie  geometrii^cil  proaression  -  +  ,^  r--  H f-T^  ^^' 

1      /^     -1     H      1  o 

Let.  therefore,  a  —  \  and  r~2\   then  will 

But  the  successive  sums  of  1,  2,  3,  4,  &c.  terms  of  this 
scries  are,       ^    (r  -i)X«_,^     MX-— 

•    (r  _  i)Xl        '^  '        T—\ 

(r^— l)Xa_  1  a 

(r^lp<«  =  r._iw-^ 
^•(r-.i)Xr2     ^        r^^     r~l 

*'(r-l)Xr'     ^        r''''      r-1 
&c.  &c. 


250     SUMMATION  OF  INFINITE  SERIES. 

Therefore 
n  terms  of  r-f-r-fr+^4'^  &c. 

s= -X       ~n  terms  of--|— {-— -f — &c. 

r — 1  \      r     r^       r^ 

These  being  the  two  series  derived  from  the   above  c\ 

pressions  ; 

But  r-^r-\-r-{-r-^-r-]-r  &c.=Mr. 


r     v 


(r-^.l), 


Whence 


a 


ed. 


2n-l 


sum  requir. 


10.  Required  the  sum  (s)    of  the  infinite   series  of  the  ^ 

reciprocals   of  the    triangular   number  t+q+-+ 1 — 

&c. 

Let  --f--4--_| — -  &c.  ad  infinitum  =s. 
1      J      tj      10 

Or, 


1,1,1,1     1  ,  '  ,  1  » 

T+2-^3  +  4  +  5+6+7^^- 

1      I       >      1       1      1    „ 

&c. 


2     3     4     5     6      7 


si 
Whence  ^~-  ;  or  s=2=  sum  required. 
-^     1 
11.  And  if  it  be  required  to  find  the  sum  of  n  terms  of 

the  same  series,  --{ \---^t7C'^t^  ^c. 

1      3      6      10      15 

Let.=l+l+i4-J+i&e.,ol, 


SUMMATION  OF  INFINITE  SERIES.     251 

Then.-144-^+^+l..c.tol. 

Therefore! L.=l4.1+ J_4.JL  &c.  toi L. . 

Or  -i-+,^+!4-4+^  &c.  to        ^ 


n+I      2     6  '  12  '  20  n(M-fl) 

Whence  ^^UU^l  &c.  to  ^- . 
n  -  1       1      d     6      lU  n(n-f-l) 

Or  T+;TH-7;+T7i+-r^  &c.  to  -_1__==-1^  =  sum  of  ?i 
1      J     6      10      15  n{n-{-\)     n-j-l 

terms  of  the  series,  as  was  required. 

12.  Required  the  sum  of  the  infinite  series ! — 

^  1.2.3     2.3.4 

Let  ■2'=Y+^-}---i—+^  &c.  ad  infinitum. 

1       ,4       o      4       O 

Then  z  —  = — [---I f-^  &c.  by  transposition. 

12     3     4     5 

^"d  ^=77^+2:3+3-+^  &c.  by  subtraction. 

^^  ^"2'^ 2:3 "^3:4'^ 475"^ 5:6  ^^*  ^^  transposition. 
Whence  l=_l-+-|-^+-.-^^  &c.  by  subtraction. 

12  2  2  2 

Or-=-^ — I — —4.— — I — —  &c. 
2     1.2.3^2.3.4^3.4.5^4.5.6 

Kut  1^2=1;  therefore --Ij+^^+^^+ji-^ 

«ii  infinitum,  ■=-,  which  is  the  sum  required. 


.^52     SUMMATION  OF  INFINITE  SERIES. 

13.  And  if  it  were  required  to  find  the  sum  of  n  terms 
of  the  same  .eries  ^-i-  +  _l_+^^+_l-g  &c. 

2^{n+i)(n+2)     2.3     3.4^4.5^5.6^6.7^ 

--  &c.  continued  to  r—.-,.,    ■,    ,r  terms. 
7.8  (n+l)(n  +  ii) 

Therefore  i-^;;;-pj-l^-  =  jJ-^+^-|-^+^^  &c. 

to  n  terms,  by  subtraction. 

Whence  l_^_^-i-— =  _i_^+^+^^  &c.   to 

M  terms,  by  division. 

And  consequently        Q'^t7T--:+.>        &c.  continued  t# 
J..C.O      ^.>5.4      0.4.5 

n  terms  =7  — ^-7— tTw— rT,T=  sum  required. 

14.  Required  the  sum  (s)  of  the  series  ^  —  t+q ^  + 

&c.  continued  ad  infinitum. 

32  -^ 

Let  x=-  and  s= — j — . 
■4  1  -j-a; 

Then  -^^xil—x+x'—x^+x'  &c.) 

And  z=(l+a:)X(a;-2^+x3-a;^+.r^  &c.) 
Whence,  by  multiplication, 
X — x^-{-x^'—x^-\-x''  &c. 


+a;2— x=^4-x^-a:5  &c. 
Whose  sum  is  =a;+0+0+0-f  0  &c. 


SUMMATION  OF  INFINITE  SERIES.     253 

X 

Therefore   z=^x,   and   x — x^-^-x^—x'^-^-x^  &c.  = — ; — 

1-rx* 

15.  Required  the  sum  of  the  series  -  +  — I 1 h&c 

2     4     8      16 
continued  ad  in/initum. 

Let  x=-  and  s= 


(i~xy 


Then r-=a;+2x2+3a;^+4a;^+5x5  &c. 

And  2=(l-.a:)2X(x+2a;24-3.r='+4x44-5x5  &c.) 
Whence,  by  multiplication, 
:f+2x^4-3x^4-4a;^  &C. 
1--2X  +  X- 


x+2x-2+3a,-='+4a;^  &c. 

~2a;2-4a;''— 6x^  &c. 

+x=^+2x*  &c. 


Whose  sum  is  =.t+04-0+04-o  &c. 
Therefore  ^=x, 

And  a;-J-2a;2+ 5x^+4x^4- 5.1^  «fec.=^^— , 

^^^-^H-^l^+S-^^^^- =(13-rp=2=sumofthe 
infinite  series  required. 

16.  It  is  required  to  find  the  sum  (s)  of  the  aeries  ^+ 

o 

4  .    9       16  ,   25    ,  .        ,     , .   ^   . 

j+^+ET  '  97q  ^^'  contmued  ad  tn/imhim. 

Let  x=--  and  ; r.=s. 

3  H-xf 

Then  7-^-=x4-4x2+9x2-|-16V+25x5  &c. 

0—xy 

\nd  2r=(l-x)3X(x+4x'2-f-9a:3+lfia;*  &c.)=x+a;-, 
as  will  be  found  by  actual  multiplication. 

^  -  i     ' 


25i     SUMMATION  OF  INFINITE  SERIES^ 

Therefore  x+a;"~z, 
And  a;+4x=+9a:=^-f  I6.r^&c.  -f-^^-^l 


Or 


(1 


14,    9    ,  16   .  -1(1+1)     3      ., 

~-i — ~-\ <^c-  =7 f^=-=I^=sum  required, 

3^9^17^81  {^-^y      2        -  ^ 

(I      a-\-d 
17.  Required   the    sum    (s)    of  the  series  — { (- 

m        mr 

a+2fZ  ,  a4-3'i  „  ,.        ,      ,  .  ^    . 

— -j —  &c.  continued  ad  infinitum. 

1 
Let  0;=-,  and  s 


r  m(l— x)" 

Then  -— t-.— + 1 ~-t 7-  &c. 

/w(l— a;)"     m        ?7ir         7«7-"  mr-^ 

Or, -| oT==a  + + — 5 V T-  &c. 

( 1  —  x")  r  H  r^ 

That  is,       ^ 


And   2=(l-x/X  Ja-i-(a+£/)5;4-(a  +  2d)x^-{-{cL^^d^. 

&c.  ;=(l~a:)a  +  ^a:, 
as  will  appear  by  actually  multiplying  by  (1  -  a-)- 

Therefore  2  =  ( 1  —  x)a  -\-  dx  ;  and   consequently  — r 

in 

a-\-d  .  a4-2(i 

mr 

nite  series  required. 

EXAMPLES. 

1.  Required  the  sum  of  100  terms  of  the  series  '2,  5,  S, 
il,  14,  &c.  Ans.  16050. 

£•.  Required    the  sum   of  50  terms  of  the  series  1+2^    ■ 
f  32^42 _|.r,2  ^^  ^j^g^  42925. 

3.  It  is  required  to  find  the  sum  of  the  series  l-f-3x-f 


1 5-  &c.  =--  {  — ^-  \  ~  sum  of  the  infi- 

mr^  m  i       (r—  \  f      5 


SUMMATION  OF  INFINITE  SERIES.     25h 

^x^-r  lOx^-{- \  5x\  contiauQd  ad  infiniium,  dec.   wfcen  x  is 
iess  than  1 . 

4,  It  is  required  to  find   the  sum  of  the   series    l-l-4x 
|- 10x^+20x^+350:",  &c.  continued  ad  infinitum^  when  x 

is  less  than  1. 

5.  It  is  required  to  find  the  sum   of  the  infinite  series 

1111  5          1 

_-- f. { j &c.  Ans.  — ,  or-. 

1.3      3.5^0.7^7.9  10'       2 

6»  Required  the  sum  of  40  terms  of  the  series  (1  X2) 

~|-(3X4)  +  (5X6)  +  (7Xe')  &c.  Ans.   86884. 

2x— I 
7.  Required    the   sum  of  n  terms   of  the   series 


2x 

7  /2x  — 

-, &r..  Ans.  nff  

2x  2x      '      2x 


2x-3  ,  2x-5     2x-7  _  .  /2x-n\ 

+— — -+-7T— +-77—  &c.  Ans.  n  y-^j  • 

8.  Required  the  sum  of  the  infinite   series   7-;r-o"T+ 

9.  Required  the  sum  of  the  series  T4-TT'77T+?r:"»"trs> 

^  1      4      10     20     cJo 

3  1 

&c.  continued  ad  infinitum.  Ans.  -,  or  1-. 

10.  It  is  required  to  find  the  sum  of  the  n  terms  of  series 

l+8x  +  27x2+64xM-)29x\  &c.  continued  ad  infinitum, 

l+4x+x2 
Ans.  — - —  —4- . 

1      2 

11.  Required  the  sum  of  n  terms  of  the  scries  — 1 — 3-+ 

3,4,5      6  1  1    ^nr+r-I> 


256     SUMMATION  OF  INFINITE  SERIES. 

12.  Required  the  sura  of  the  series 1 1 

J — L_  &c.   .  .   .  A ? * 

^8.12  ^2n(;i+2n)* 

.         ^       3  5n-\-Sn- 


16'        32+4Sn+l6n- 
13.  Required  the  sum  of  the   series  r-^~l" 


+.^&c... 


J2.20  3u{4-\-4n) 


3.8      6.12   '    9.1b- 


Ans.  2  =  -—,  s: 


12'        124-i2tJ 

14.  Required  the  sum  of  the  series 1 — H 

^  2.7       7.12        12.17 


17.22  •  (5n_3).(6n-f-2) 

8  3« 

Ans.  2=-,ii— --—--, 
5  2-i-5« 

15.  Required  the  sum  of  the  scries  — f-- 


3.6      6.8      9.10 


'         •     &C.    .    .    .    ±:  ' 


12.12    '  Sn{4-i-2p) 

" ii*  ^ "" 2(3+6^0' "~ 4(6 +  6n)  - 
2  3  4 

16.  Required  the  sum  of  the    series 1 -  — 

^  3.6      6.7        7.9 

^-  &c.  .   .   .   -^ — ■■ ^ 

9.11    '  ~(l+2«).(3  +  2n) 

Ans.  2  =  —-,  s=--  — 


1.  12     4(34-4n) 


*  The  symbol  Z,  made  use  of  in  these,  and  some  of  ihe  following  series 
denotes  the  sum  of  an  infinite  number  of  terms,  and  S  the  sum  of  n  term? 


LOGARITHMS.  257 

3  6 

17.  Required   the  sum   of  the   series  t-^-^-I-k'^^'^ 


.2.3  •  2.3.4 


7,8  .  4-fn 

4--~— &e. 


3.4.5  '  4.5.6  n(l+n).(2+n) 

^3         3        2,1. 

Ans.2=-,s=^-.^-p^+2^*. 

Of  logarithms. 

Logarithms  are  a  set  of  numbers  that  have  been  com- 
puted and  formed  into  tables,  for  the  purpose  offacilitat 
ing  many  difficult  arithmetical  calculations ;  being  so 
contrived,  that  the  addition  and  subtraction  of  them  answers 
to  the  multiplication  and  division  of  natural  numbers  with 
which  they  are  made  to  correspond!. 


*  The  series  here  treated  of  are  such  as  are  usually  called  algebraical) 
which,  of  course,  embrace  only  a  small  part  of  the  whole  doctrine.  Those, 
iherefore,  who  may  wish  for  iariher  informjition  on  this  abstruse  but  hij^hly 
:ijrious  subject,  are  referred  to  the  Miscellanea  Analyticaof  Demoivre,  Ster- 
ling's Jllethod  Dijfer.,  Jame-;  Beniouilli,  de  Seri.  Infin.,  Simpson's  Math 
Dissert.  Waring's  Medii.  Analyt.^ClarVs  translation  oi Lorgna's  Series,  tht 
various  works  of  Euler,  and  Lacro'x  IVaite  du  Calcul  Diff.  et  Int.,  where 
they  will  find  nearly  a'l  the  materials  thai  have  been  hitherto  collected  re- 
specting tiiii  branch  of  analysis, 

f  This  mode  of  compulation,  which  is  one  of  the  happiest  and  most  use- 
ful discoveries  of  modern  times,  is  due  to  Lord  Napier,  Baron  of  Merchiston, 
in  Scotland,  who  first  published  a  table  of  these  numbers,  in  the  3  ear  1614, 
under  the  title  o{  Canoti  Mirijicum  Logarithmorum  ;  which  perforjnance 
was  eagerly  received  by  the  It  arned  throughout  Europe,  whose  efforts  were 
immediately  directed  to  the  improvement  andextensionsof  the  new  calculus, 
that  had  so  unexpectedly  presented  itself. 

Mr.  Plenry  Briggs,  in  particular,  who  was,  at  that  time,  professor  of  geo- 
metry in  Gresham  College,  greatly  contributed  to  the  advancement  of  thi? 
doctjine,  not  only  by  the  very  advantageous  alteration  which  he  first  introduc- 
ed into  the  system  of  I'nese  numbers,  by  making  1  the  logarithm  of  10,  instead 
of  2.3U25852,  as  had  been  done  by  Napier  ;  but  also  by  the  publication,  in 
1624  and  1633,  of  his  two  great  works,  the  Arithmeiica  Logarithmica  and 
the  Trigonometria  Brilanica,  both  of  which  were  formed  upon  the  principle 
above  mentioned  :  as  are,  likewise,  all  our  comnon  logarithmic  tables  a*^. 
orescnt  in  use. 


258  LOGARITHMS. 

Or,  when  taken  in   a    similar   but    more  general  sensr 
logarithms    may  be   considered  as   the   exponents  of  the 
powers   to  which   a  given   or   invariable  number  must  be 
raised,    in    order  to  produce  all   the  common,  or  natural 
numbers.     Thus,  if 

a^=:?/,  a""'  -  y ,  a''"=-y'\  &c. 
then  will  the   indices  a-,  a',  x'\  &c.  of  the    several  powers 
of  a,  be  the  logarithms  of  the  numbers?/,  y  ,  y\  8zc.  in  the 
scale,  or  system,  of  which  a  is  the  base. 

So  that,  from  either  of  these  formulae  it  appears,  that 
the  logarithm  of  any  number,  taken  separately,  is  the  in- 
dex of  that  power  of  some  other  number,  whirh,  when  in- 
volved in  the  usual  way,  is  equal  to  the  given  number. 

And  since  the  base  a,  in  the  above  expressions,  can  be 
assumed  of  any  value,  greater  or  less  than  1,  it  is  plain  that 
there  may  be  an  endless  variety  of  systems  of  logarithms, 
answering  to  the  same  natural  number. 
,  It  is,  hkewise,  farther  evident,  from  the  fust  of  these 
equations,  that  when  y—l,  a-  will  be  =(>,  whatever  may 
be  the  value  of  a  ;  and  consequently  the  logarithni  oi"  1  is 
always  0,  in  every  system  of  logarithms. 

And  if  a:=^l,  it  is  manifest  t>oni  the  same  equation,  that 
the  base  a  will  be  =?/;  which  base  is  therefore  the  num- 
ber whose  proper  logarithm,  in  the  system  to  ^\hif'h  if  \,v- 
longs,  is  1. 

Also,  because  a''=y,  and  a'^'—y,  it  follows  i^rom  tin 
multiplication  of  powers,  that  ff^Xa^',  or  a^t-'=z/y  ;  and 
consequently,  by  the  definition  of  logarithms,  given  above. 
;.-{-x'=log.  yy,  or 

log.  yy'^\og,y^-]o'^.  y-     ^   ^ 
And,  for  a  like  reason,  if  any  nuniber  of  the  equations 
,/^  =  i/,  a'''=y',  a''"=y",  &c.  be  mullipliod  together,  we  shall 
iiavea'^'l'^+^'  &.c.'=yy'y'\  d:c.  ;  and  consequently  a; -{-x'-f-^'r 
vS:c.  —log.  yy'y",  &c.  ;  or 

log-  yy'y"  ^^- ;  =iog-  2/-r  log-  y+  log-  y"  <^c. 

See,  for  farther  details  on  this  part  of  the  subject,  the  Intioduction  to  mv 
j'reaiise  of  Plane  and  Spherical  Trigonometry,  8vo.  2d.  Edit.  1813  ;  and  for 
fiie  construction  and  use  of  the  tables  consult  thoseof  Sherwin,  Hutton,  Tay- 
lv>r,  Callet,  and  Borda,  where  ever  necessary  information  of  this  kind  may 
be  readily  obtained. 


LOGARITHMS.  259 

From  which  it  is  evident,  that  tlie  logarithm  of  the  pro- 
duct of  any  number  of  factors  is  equal  to  the  sum  of  the 
logarithms  of  those  factors. 

Hence,  if  all  the  factors  of  a  given  number,  in  any  case 
of  this  kind,  be  supposed  equal  to  eacii  other,  and  the  sum 
of  them  be  denoted  by  in,  the  preceding  property  will 
then  become 

log.  ?/'»=m  log.  y. 

From  which  it  appears,  that  the  logarithm  of  the  wth 
power  of  any  number  is  equal  to  m  times  the  logarithm  of 
that. number. 

In  like  manner,  if  the  equation  a''-=y  be  divided  by  a' 
=^y' ,  v/e  shall  have,  from  the   nature  of  powers,  as  before, 

— ,  or  «^-'''  =  -,  ;  and  by  the   defmition  of  logarithms,  laid 
a"-  y 

y 

down  in  the  first  part  of  this  article,  a:  —  a''  =  log.— ,  or 

y 

log.--=log.  2/  — log.  ^'. 

Iience  the  logarithm  of  a  fraction,  or  of  the  quotient 
arising  from  dividing  one  number  by  another,  is  equal  to 
the  logarithm  of  the  numerator  minus  the  logarithm  of  the 
denominator. 

And  if  each  member  of  the  common  equation  a''=^y  be 

raised  to  the  fractional  power  denoted  by  — .  we  shall   have 

n  ' 

in  that  case,  a**   =?/"  ; 

And,  consequently,  by  taking  the   logaritlmis,  as  before. 

7rt  in 

-a:=log.  ?/",  orlog.  f  =—  log.  y. 

Where  it  appears,  that  the  logarithm  of  a  mixed  root, 
or  power,  of  any  number,  is  found  by  multiplying  the  lo- 
garithm of  the  given  number  by  the  numerator  of  the  in- 
dex of  that  power,  and  dividing  the  result  by  the  denomi- 
nator. 

And  if  the  numerator  m,  of  the  ilactional  index,  be  in 
this  case,  taken  equal  to  1,  the  above  formula  will  then 
become 


260  LOGARITHMS. 

From  which  it  follows,  that  the  logarithm  of  the  ?ai: 
root  of  any  number  is  equal  to  the  nth  part  of  the  loga- 
rithm  of  that  nufnber. 

Hence,  besides  the  use  of  logarithms,  in  abridging  the 
operations  of  multiplication  and  divi-^ion,  they  are  equally 
applicable  to  the  raising  of  powers  and  extracting  of  roots  ; 
which  are  performed  by  simply  multiplying  the  given  lo' 
garithm  by  the  index  of  the  power,  or  dividing  it  by  thQ 
number  denoting  the  root. 

But  although  the  properties  here  mentioned  are  com- 
mon to  every  system  of  logarithms,  it  was  necessary,  for 
practical  purposes,  to  select  some  one  of  them  from  the 
rest,  and  to  adapt  the  logarithms  of  all  the  natural  numbers 
to  that  particular  scale. 

And,  as  10  is  the  base  of  our  present  system  of  arithme 
tic,  the  same  number  has  ac'-ordmsly  been  chosen  for  the 
base  of  the  logarithmic  system,  now  generally  used. 

So  that,  according  to  this   scale,  which   is   that  of  the 
common  logarithmic  tables,  the  numbers 
.  .   .    ]0-\  lO-\  10-^  lO-S  10",  lOS  10-,  10\  IU\  kc. 

Or 

•  •  •  Tom^T^o^m  ^'  ^^'  ''''  ''''^  '''''^  ^^' 

have  for  their  logarithms 
....  — 1,— 3,  —2,  -1,  0,  1,  '2,  3,  4,  &c. 

Which  are  evidently  a  set  of  numbers  in  arilhmelical 
progression,  answering  to  another  set  in  geometrical  pro- 
'>ression  ;  as  is  the  case  in  every  system  of  logarithms. 

And  therefore,  since  the  common  or  tabular  logarithm 
of  any  number  (n)  is  the  index  of  that  power  of  10,  which^ 
when  involved,  is  equal  to  the  given  number,  it  is  plain 
from  the  following  equation, 

l(y  =  n,  or  10-^=-, 

?i 

that  the  logarithms  of  all  the  intermediate  numbers,  in  the 
above  series,  may  be  assigned  by  approximation,  and  made 
H  occupy  their  proper  places  in  the  general  scale. 


LOGARITHMS.  26i 

it  IS  also  evident,  that  the  lo|rarithms  of  1,  lO,  100^ 
iOOO,  &c.  beinij;  0,  1,  2,  8,  &;c.  respectively,  the  logarithm 
of  any  number,  falling  between  0  and  I,  will  be  0  and 
some  decimal  pirts;  that  of  a  number  between  10  and  100, 
1  and  somn  decimal  parts  ;  of  a  nu.nber  between  lOU  and 
1000,  2  and  some  decimal  parts  ;  and  so  on,  for  other  num- 
bers of  this  kind. 

And,  fur  a   similar    reason,  the  losiarithms  of  — ,  —rr. 
'  "^^  10    100 

,  &c.  or  of  their  equals  .1,  .Oi,  .001,  &c.  in   the   de- 

1000'  ^ 

scendin^  part  of  the   scale,  bpins  — ',  — 2,  — 3,  &c.  the 

logarithm  of  any  number,  falling    between  0  and  I,  will  be 

—  1,  and  some    positive   det*imal  parts  ;  that    of  a  number 

between  .1  and  .01,  — i^,  and  some  positive  decimal  parts  ; 

of  a  number  between  .01  and    001,  — r^,  and  some  positive 

decimal  parts  ;  &c. 

Hence,  likewise,  as  the  muhiplyino;  or  dividing  of  any 
number  by  10,  h'O,  *000,  &c,  i-  performed  by  barely  in- 
creasing or  diminishing  the  integral  part  of  its  logarithm 
by  I,  2,  ^,  &c.  It  i-i  oUvious  that  ail  numbers,  which  con- 
sist of  the  sine  Si^ures.  wheth<.'r  they  be  integral,  frac- 
tional, or  mixed,  will  have,  for  the  decimal  part  of  their 
logarithms,  the  sam^  positive  quantity. 

So  that,  in  this  svstem,  tlie  integral  partolany  logarithm, 
which  is  usually  calh;d  its  index,  or  characteristic,  is  al 
ways  hess  by  *  than  the  number  ol"  iriteiiers  which  the  natu- 
ral number  consists  of ;  and  f  )r  decimals,  it  is  the  number 
which  denotes  the  distance  of  tlio  rtrst  significant  figure 
from  the  place  of  units. 

Thus,  according  {>  the  logarithmic  tables  in  common 
use,  wc  have 


\''u  fibers. 

Loffitrilhtus. 

1.3-HiU 

0.1361496 

20  05(t0 

'.;>0  *s  144 

33o.yb0 

2.52o;3S17 

.^b52i 

i.fi676490 

.0    154 

^.7-^91. 57  fi 

&c. 

<^'C, 

262  LOGARITHMS. 

Where  the  sign  —  is  put   over  the  index,  instead  of  be 
fore  it,   when  that  part  ot  the  logarithm  js  neuaiive,  in  or- 
der to  distinguish  it  from  the  decimal  part,  which  is  alway- 
to  be  considered  as  -p,  or  affirmative. 

Also,  agreeably  to  what  has  been  before  observed,  the 
logarithm  of  3.^540  hem;:  4.5859117,  the  loijafilhms  of 
any  other  numbers,  consisimg  of  the  same  ngures,  will  be 
as  follows : 

Numbem.  .  Logarithms. 
.^S54     :3.5859i  i7 


3«5. 4  '  2...85M  17 


38.54 

3.-54 

.3S54 

.03'  54 

.003.^54 


1.5-59117 

0.585y ( I  7 
T5^5.^  17 
^.58591  17 
3.58591 17 


Which  logarithms,  m  this  case,  as  well  as  in  all  others 
of  a  similar  kind,  whether  the  number  contains  ciphers  or 
not,  differ  only  in  their  indices  the  decimal,  or  positive 
part,  being  the  .same  in  them  all.* 

And,  as  the  indice-,  or  integral  parts,  of  the  logarithms 
of  any  numbers  whatever,  m  this  system,  can  always  be 
thus  readily  found  from  the  simple  consideration  of  the 
rule  above  mentioned,  thev  are  generally  omitted  in  the 
tables,  being  left  to  be  supplied  by  the  operator,  as  occa- 
sion requires. 


*  The  great  advatilagfcs  attefidiiis  the  common,  or  Brig°ean  system  of  lo 
garithms,  above  all  othfrs,  arisf  chiefly  (ion\  the  readiiies>  with  which  we 
can  ahvaysfind  the  chaiHcteri.*ti<-  -r  ititet^ral  part  of  any  lofcarithm  from  the 
bare  inspection  of  the  natural  number  to  whicli  it  belongs,  and  the  circum 
stance,  that  muhiplyinj;  or  dividing  any  immber  b)'!  :,  100,  lOOO  &,<j.  only 
influences  the  characteristic  o:  its  lo<iari:hm,  without  afft^ting  the  decimal 
part.  Thus,  for  instance,  if  i  be  made  to  denote  the  inaex  or  integral  part  ot 
ihe  logarithm  of  any  number  n,  and  d  its  decimal  pait,  we  shall  have  log,  > 

=r>t-t-(i;  log.  lO^'x  N  =  (z4-jn)-|-rf;  log. -^ ■=  (i— m)-}- ci ;  where  it  i? 

plain  that  the  deci.nal  part  of  the  log-anthm.  in  each  of  these  cases,  remaif'-s 
ihe  same. 


LOGARITHMS.  26^ 

It  may  here,  also,  be  farther  added,  that  when  the  loga- 
rithm of  a  given  number  in  any  particular  syj^iem,  isknown^ 
it  will  be  easy  to  tind  the  logarithm  of  the   same  number  in 
any  other  system,  by  means  of  the  following  equations, 
a^=w,  and  e^'=n.  or  log.  »=rr,  and  /.  n  =  x'. 

Where  log.  denotes  the  logarithm  of  7a  m  the  system  of 
which  a  is  the  base,  and  I.  its  logarithm  m  the  system  of 
which  e  is  the  base. 

For,  since  rt^^=e^',  or  aa^'=c,  and  e^=a,    we  shall  have, 

X 

for  the  base  a,  -7=  log.  e,  or  x=x'  log.  e  ; 

x' 

and  for  the  base  e,  -=/.  a,  or  x'=x  I,  c. 

:r 

Whence,  by  substitution,  from  the  former  equations^ 
log.  n=U  nXlog.  e;  or  log.  n  =  /.  71 X—, 

Where  the  multiplier  log.  e,  or   its  equal  — ,    expresses 

the  constant  relation  which  the  logarithms  of  n  have  to 
each  other  in  the  systems  to  which  they  belong. 

But  the  only  system  of  these  numbers,  deserving  of  no- 
tice, except  that  above  described,  is  the  one  that  furnishes 
what  have  been  usually  called  hyperbolic  or  Neperian  lo- 
garithms, the  base  e  of  which  is  2.' 1828l!S28459  .   .   . 

Hence,  in  connparing  these  vvjth  the  common  or  tabular 
logarithms,  we  shall  have,  by  putting  a  in  the  latter  of  the 
above  formultG  =10,  the  expression, 

log.  71=/.  nXi— -,  or  /.  ?i  =  Iog.  nXl.  10. 

Where  log.  in  this  case,  denotes  the  common  tabular 
logarithm    of  the   number  n,  and   /.   its  hyperbolic   loga- 

•■ilhm ;  the   constant   factor,  or    multiplier,-^ — ,  which    is 


264  LOGARITHMS. 

bein^  what  is  usually    called  llie  modulus  of  the  commoii 
system  of  logarithms.* 

PROBLr  M  I. 

To  compute  the  logarithm  of  any  of  the  natural  num- 
bers, 1,  2,  3,  4,  5,  ^c. 

RULE  I. 

1.  Take  the  geometrical  sories,  1,  10,  100,  1000,  10000, 
&c,  and  apply  to  it  he  ariihmetical  series,!),  1,2,3,4, 
«kc.  as  logarithms. 

2.  Find  a  geometric  mean  ht^tvveen  1  and  10,  10  and 
100,  or  any  other  two  fidjarent  terms  of  the  series,  be- 
twixt which  the  number  proposed  lies. 

3.  Also,  between  the  mean,  thus  found,  and  the  near^ 
est  extreme,  find  another  ge^  metrical  mean,  in  the  same 
manner  ;  and  so  on,  till  you  are  arrived  within  the  pro- 
posed limit  of  the  number  whose  logarithm  is  sought. 

4.  Find,  likewise,  as  m.any  arithmetical  means  between 
the  corresponding  terms  ui'  the  other  seiies,  0,  1,2,  3,  4, 
&:c.  in  the  same  order  as  you  iound  the  geometrical  ones, 
and  the  last  of  these  will  be  the  logarithm  answering  to  the 
number  required. 

EXAMPLES. 

Let  it  be  required  to  find  the  logarithm  of  9. 


*  It  may  here  be  remarked,  that  although  the  common  logarithms  have 
superseded  the  use  of  hyperbolic  or  IVepenan  logarithms,  in  all  the  ordinary 
operations  to  which  these  numbers  are  generally  applied,  yet  the  latter  are 
not  without  some  advantages  peculiar  to  themselves*,  being  of  frequent  oc- 
i-urrence  in  the  application  cf  (he  Fluxionary  Calculus,  to  many  analytical 
and  physical  problem?,  where  they  are  required  for  ihe  finding  of  certain  flu^ 
ents,  which  could  not  be  so  readily  determined  without  their  assistance  ;  on 
which  account  great  pains  have  been  taken  to  calculate  tables  of  hypeibolic 
logarithms,  to  a  considerable  extent,  chiefly  for  this  purpose.-  Mr.  Barlow, 
in  a  Collection  of  Mathematical  Tables  lately  published,  has  given  them  fo! 
rhe  first  10000  numbers. 


LOGARITHMS.  265 

Here  Ihe  proposed  number  lies  between  1  and  10. 

First,  then,  the  lo^.  of  lU  is  I,  and  the  Iojj.  of  1  is  0. 

Therefore    ^^(10  >^  1)=  i/ '0=3.162-777    is  the  geo- 
metrical mean  ; 

And  1(1-1-0)=^  =  .5  is  the  arithmetical  mean; 

Hence  the  log.  of  3.162i''777  is  .5 

Secondly  the  lo^.  of  10  is  1,  and  the  log.  of  3.1622777 
i3  .5. 

Therefore  v(  »0X3.  J622777)=5  b234la2  is  the  geo- 
metrical mean  ; 

And  i(14-.5)  =  .75  is  the  arilhmetjcnl  mean  ; 

Hence  the  Iojj.  of  5.62:^132  is  .75. 

Thirdly,  the  |..g.  of  10  is  1,  and  the  log.,  of  5.6234132 
is  .75  ; 

Therefore  v/(  1^X^-62:^4 1 32)  =7.4989422  is  the  geo- 
metrical nteaf!  ; 

And  J  (I  -f-. 75)  =  . 875  is  the  arithmetical  mean; 

Hence  the  log.  of  7.498^)422  is  .875. 

Fourthly,  the  log.  of  lu  is  1,  and  the  log.  of  7.4989422 
is  .875  ; 

Therefore    v/(10X7.4989422)=S.6596431  is  the  geo- 
metrical  mean  ; 

And  i(H-. 875)=. 9375  is  the  arithmetical  mean; 

Hence  the  log.  of  8.(S59643l  is  .9375. 

Fifthly,  the  log.  of  10  is  1,  and  the  log.  of  8.6596431  is 
,9375. 

Therefore  ^/(lO X8.6596431)  =  9.3057204  is  the  geo- 
metrical mean. 

And  A (1 +.9375)  =  .96875  is  the  arithmetical  mean  ; 

Hence  the  log.  of  9.3057204  is  .96875. 

Sixthly,  the  log.  of  8.6696131  is  .9375,  and  the  log.  of 
9.3057204  is  .96875  ; 

Therefore    V  (8-6596431  X9.3057204)=8.9768713  is 
the  geometrical  mean. 

And   i(.9375+.96875)=.963125  is    the  arithmetical 
mean  ; 

Hence  the  log.  of  8.9768713  is  .963125. 

And,  by  proceeding  in  this  manner,  it  will  be  found,  after 
25  extractions,  that  the  logarithm  of  8.9999998  is  .9542425, 

A  a 


'2QQ  LOGARITHMS. 

which  may  be  taken  for  the  logarithm  of  9,  as  it  difFeis 
from  it  so  little,  that  it  may  be  considered  as  sufficiently 
exact  for  all  practical  purposes. 

And  in  this  manner  were  the  logarithms  of  all  the  prime 
numbers  at  first  computed. 

RULE  II. 

When  the  logarithm  of  any  number  («)  is  known,  the 
logarithm  of  the  next  greater  number  may  be  readily 
found  from  the  following  series,  by  calculating  a  sufficient 
number  of  its  terms,  and  then  adding  the  given  logarithm 
to  their  sum. 

Log.  (n  +  l)=log.  n  +  M'  j  ^-4-_L_3+_l^^^^ 

+._! +_J___i. I &c   f 

Or, 

Log.  (.+  l)-log.  ni-\^-^^'-~+^^^. 

7{2n+iy  ^9(2?i+l)-      ll(2n+])^        " 

Where  a,  e,  c,  &c.  represent  the  terms  immediately 
preceding  those  in  which  they  are  first  used,  and  m'= 
twice  the  modulus  =.8685889638  ....  * 

EXAMPLES. 

L  Let  it  be  required  to  find  the  common  logarithm  of 
the  number  2. 

Here,  because  n+l=2,  and  consequently  ?i=l  and  ti 
+  1=3,  we  shall  have 


*  It  may  here  be  remarked,  that  the  difference  between  the  logarithms  u." 
iny  two  consecutive  numbers,  is  so  much  the  less  as  the  numbers  are  great- 
v\  and  fconsequently  the  series  which  comprises  the  latter  part  of  the  above 
;xpreSbiou  will  in  that  case  converge  so  much  the  faster.     Thus  log.  u  and 

log.  {ii'\-1),  or  its  equal  log.  n-|-log.  (l-f. -),  will,  obviously,  differ  but  lit 

le  from  each  other  when  n  is  a  large  number. 


LOGARITHMS.  267 

M'         8685889638      ^  .289529654  (a) 


2n4-  I      3 


2895.9654   ^  ,010723321  (b) 


3(2n-fl)2      3.32 
__^B_^3X.01^723321^  .000714888  (c) 
5(2n+l)2       5.32 

5c    _^X.000714888^^^^^^^g^3^^^, 


7(2n-l-l)^       7.3^ 

__J^_=L^^.O-^'i^^£!?^=  .000004903  (E) 
9(2n+l)2       9.3- 

9e     9X.OuO004903_^^^^^^^^^^^.^^, 


Il(2n4-1)2      11.32 

_JlL__.=il^^l^«il5=  .000000042  (o) 
13(2w+0'       '•'^•3' 

.30     '3X.0OO000042^^^^^^^^^^(^^ 


15(2n-flf      15.2^ 


Sum  of  8  terms  .  .  .30lu29995 
Add  log.  of  1   .  .  .000000000 


Log.  of  2   ...  .301029995 

Which  logarithm  is  true  to  thn  last  figure  inclusively. 

2.   Let  it  be   required    to  coiapute  the  logarithm  of  the 
number  3. 

Here,  since  n-}-l=3,  and  consequently   ?i=2,  and  2n 
+  1=5,  we  shall  have 

_m;__^^58_896_4  ^  ^  ^  ^.173717798  (a) 

^ =  •22^'^  Ml^^  ,  .  .  =.002316237  (b) 

3(2n  +  l)2  3.52 

_J-_^i_X_^2326237_^^^^^5.5^0  (c) 
5(2n4-l)2  5.52 

_lg__-,^'QQQ^^^^QQ  =.000001588  (D) 


7(2n-{-J)2  7.5== 


268  LOGARITHMS. 

7d  7X.00()0(U5S8 

9e  9X.O0U()00'.5o  ,  , 

Sum  of  6' terms      ....     .17609i260 
Log.  of  2 .t)10y99.-t5 

Log.  of  3 477121255 

Which  logarithm  is  also  correct  to  the  nearest  unit  n-^ 
the  last  fifiure. 

And  m  the  same  way  we  may  proceed  to  find  the  loga* 
rithm  of  any  prinie  number. 

Also,  because  thf^  sum  of  the  logarithms  of  any  two  num- 
bers gives  the  logarithm  of  their  product,  and  the  differeuco 
of  the  lugaritlims  the  U)gariihm  of  their  quotient,  &c.  ;  we 
may  readily  compute,  from  the  above  two  logarithms,  and 
the  logarithm  ai  lu,  which  is  1,  a  great  number  of  other 
logarithms,  as  in  the  following  examples  : 

3.  Because  2Xt>-4,  therefore  i       .3,^029995 

log.  ^J  § 

Mult,  by  2  2 

gives  log.  4  .60^0599-^0 

4.  Because  2 X3=b,  therefore  ^        30i029^r^5 

to  lug.  2^ 
add  log.   3     .477121255 

gives  log.   6     .7781-1250 

5.  Because  2^=8,  therefore  log.  2     .301029995 

mult,  by  3  3 

gives  log.  8     .9i  3t<89985 


LOGARITHMS.  269 

6.  Because  32=9,  therefore  log.  3  .477121265 

mult,  by  2  2 

gives  log.  9  .954242510 

7.  Because  V  =5,  therefore  from  >  I.OOOOOOOOO 

log.  10  ^ 
lake  log.  2  .301029995 

gives  log.  5  .698970006 

8.  Because  3X4—12,  therefore  ^      477121265 

to  log.  3    ^     ' 
add  log.  4  .602059991 


gives  log.  12      1.0791S1246 

And,  thus,  by  computing,  according  to  the  general  for- 
mula, the  logarithms  of  the  next  succeeding  prime  num- 
bers 7,  11,  13,  J.  7,  19,  23,  &c.  we  can  find,  by  means  of 
the  simple  rules,  before  laid  down  for  multiplication,  divi- 
sion, and  the  raising  of  powers,  as  many  other  logarithms 
as  we  please,  or  may  speedily  examine  any  logarithm  in 
the  table. 


MULTIPLICATION 

BY  LOGARITHMS. 

Take  out  the  logarithms  of  the  factors  from  the  table, 
and  add  them  together  ;  then  the  natural  number,  answer- 
ing to  the  sum,  will  be  the  product  required. 

Observing,  in  the  addition,  that  what  is  to  be  carried 
from  the  decimal  part  of  the  logarithms  is  always  affirma- 
tive, and  must,  therefore,  be  added  to  the  indices,  or  inte- 
gral parts,  after  the  manner  of  positive  and  negative  quan- 
tities  in  algebra. 

Aa2 


1>70  LOGARITHMS. 

Which  method  will  be  found  much  more  convenient, 
to  those  who  possess  a  sHght  knowjedje  of  this  science- 
than  that  of  using  the  arithmetical  complements. 

EXAMPLES. 

1.  Multiply  37.153  by  4.086,  by  logarithms. 
JSi'os.  Lo(:;s. 

37.   53  .....      1.5699939 
4.Ub6 0.61129f'4 


Prod.  151.8071     .      .      2.  i812923 


2.  Multiply  112.246  by  13.958,  by  logarithms. 
JVos.  Logs. 

112.246       ....      2.0491709 
13.958   .....     1.14482:32 


Prod.  1663.  r28    .     .     3.1939941 


3.  Multiply  46.7512  by  .3^75,  by  logarithms. 
JVo5.  Logs. 

46.7512       ....      1.6697928 
.3275 T.5162115 


Prod.   15.31102    .     .      1.1850041 


Here,  the  +I5  that  is  to  be  carried  from  the  decimals, 
cancels  the  —  1>  and  consequently  there  remains  1  in  the 
upper  line  to  be  set  down. 

4,  Multiply  .37S1U  by  .04782,  by  logarithms. 
J^os.  Logs, 

.37816  1.5776756 

.04782  2.679B096 


Prod.  .0180836    .     .     2.2572852 


Here  the  + 1  that  is  to  be  carried  from  the  decimals. 


LOGARITHMS.  271 

destroys  the  — 1,  in  the  upper  line,  as  before,  and  there 
remains  the  -  2  to  be  set  down. 

5.  Multiply  3.7b8,  2.U53,  and  .007693,  together. 

JVos.  Lo^s. 

7.768     ....  0.5761109 

2.('53     ....  0.3*23889 

.007693      .     .     .  3.8860997 

Prod. .05951 1       .     2.7745995 

Here  the  -fl,  that  is  to  be  carried  from  the  decimals, 
when  added  to  -  3,  makes  —2  to  be  set  down. 

6.  Multiply  3.586,  2. 1046,  .8372,  and  .0294,  together 

NoH.  Logs. 

3.586  ....  0.554610 

2.1046  ....  0,323170 

.8372  ....  1.922829 

.0294  ....  2.468347 

Prod. .1857618  .  .    1.268956 


Here  the +2,   that  is  to  be    carried,   cancels  the  --2> 
and  there  remains  the  —  I  to  be  set  down. 
7.  Multiply  23.14  by  5.062  by  logarithms. 

Ans.  117.1347. 
0.  Multiply  4.0763  by  9.8432,  by  logarithms. 

Ans.  40.12383. 

9.  Multiply  498.256  by  41.2467,  by  logarithms. 

Ans.  20551.41. 

10.  Multiply  4.026747,  by  logarithms. 

Ans.  .0497102. 

n.  Multiply  3.12567,  .028G8,  and  .12379,  together, by 
logarithms.  Ans.  .09109705. 

12.  Multiply  2876.9,  .10674,  .098762,  and  .0031598, 
by  logarithms.  Ans.  .0058299. 


(272) 


DIVISION  BY  LOGARITHMS. 

From  the  logarithm  of  the  dividend,  as  found  in  the  ta- 
bles, subtract  the  logarithm  of  the  divisor,  and  the  natural 
number  answering  to  the  remamder,  will  be  the  quotient 
required. 

Observing,  if  the  subtraction  cannot  be  made  in  the 
usual  way,  to  add,  as  in  the  former  rule,  the  I  that  is  to  be 
carried  from  the  decimal  part,  when  it  occurs,  to  the  in- 
dex of  the  logarithm  of  the  divisor,  and  thea  this  result, 
with  its  sign  changed,  to  the  remaining  index,  for  the  index 
of  the  logarithm  of  the  quotient. 

EXAMPLES. 

1.  Divide  4768.2  by  36.954,  by  logarithms. 
JYos.  Logs. 

476B.2     ....     3.6783545 
36.954     ....      1.5676615 


Quot.  129.032  .      .     2.1106930 


'2.  Divide  21.754  by  2.4678,  by  logarithms. 
JYos.  Logs. 

21.754     ....      1.3375391 
2.4678     ....     0.3923100 


Quot.  8.1518     .     .     U.9452291 

8.  Divide  4.<  257  by  .17608,  by  logarithms. 
Nos.  Logs. 

4,6257     ....      0.6651725 
.17608     ....     1.2457100 


Quot.  26.2741   .     .     1.4194625 


DIVISION  BY  LOGARITHMS.  273 

Here  —1,  in   the  lower  index,  is   changed  into    -f  1, 
which  is  then  taken  for  the  index  of  the  result. 
4.  Divide  .276ii4  by  5.1576,  hy  logarithms. 

JVo.f.  Logs. 

.27H84 r.442^2SS 

5.1676 0.7124477 


Quot.  .053^761    .      .     2.72MT8J1 


Here  the   I    that   is  to  be  carried  from  the  decimals,  is 
taken  as  — 1,  and   then  added   to  —  I,  in  the  upper  index, 
which  gjives  —2  for  the  index  of  the  result. 
5.  Divide  6.9875  by  .075789,  by  looarithms. 
Nos.  L"f^s, 

6.9875 0.844.^218 

.0767ti9       ....     2.87.^6062 


Quot.  92.1967       .     .      I.96i7156 


Here  the  1,  that  is  to  be  carried  from  the  decimals,  is 
added  to  — 2,  which  makes  —  1,  and  this  put  down,  with 
its  sign  changed,  is  4-  i. 

6.  Divide  .  19876  by  .0012345,  by  logarithms. 
JVus.  Lo^s. 

.19876 r.2983290 

.0012.345     ....     3.09  4.'11 


Quot.   161.0051    .      .      2  2069879 


Here  —3,  in  the  lower  index,  is  changed  into  -|-3,  and 
this  added  to  —  I,  the  other  index,  jjives  +3—  1  or  2. 

7.  Divide  125  by  172«,  by  logarithms. 

Ans.  0723379. 

8.  Divide  1728.95  by  1.10678,  by  logarithms. 

Ans.    1562.144. 

9.  Divide  10.23674  by  4.96523,  by  logarithms. 

Ans.  2.061685c 


274    RULE  OF  THREE  BY  LOGARITHMS. 

10.  Divide  19936.7  by  .048235,  by  logarithms. 

Ans.  .413739. 
n.  Divide  .067059  by  1234.59,  by  logarithms. 

Ans.  .0000549648. 


THE  RULE  OF  THREE, 

OR  PROPORTION,  BY  LOGARITHMS. 

For  any  single  proportion,  add  the  logarithms  of  the  se- 
cond and  third  terms  together,  and  subtract  the  logarithm 
of  the  first  from  their  sum,  according  to  the  foregoing 
rules  ;  then  the  natural  number  answering  to  the  result 
will  be  the  fourth  term  required. 

But  if  the  proportion  be  compound,  add  together  the 
logarithms  of  all  the  terms  that  are  to  be  mulfiplied,  and 
from  the  result  take  the  sum  of  the  logarithms  of  the 
other  terms,  and  the  remainder  will  be  the  logarithm  of 
the  term  sought. 

Or,  the  same  may  be  performed  more  conveniently  thus, 

Find  the  complement  of  the  logarithm  of  the  first  term 
of  the  proportion,  or  what  it  wants  of  lO,  by  beginning  at 
the  left  hand,  and  taking  each  of  its  figures  from  9,  except 
the  last  significant  figure  on  the  right,  whicii  nmst  be  taken 
from  10;  then  add  this  result  and  the  logarithms  of  the 
other  two  terms  together,  and  the  sum,  abating  10  in  the 
index,  will  be  the  loijarithm  of  the  fourth  term,  as  before. 

And,  if  two  or  more  logarithms  are  to  be  subtracted,  as 
in  the  latter  part  of  the  above  rule,  add  their  complements 
and  the  logarithms  of  the  terms  to  be  multiplied  together, 
and  the  result,  abating  as  many  10's  in  the  index  as  there 
are  logarithms  to  be  subtracted,  will  be  the  logarithm  of 
the  term  required  ;  observing  when  the  index  of  the  lo- 
garithm, whose  complement  is  to  be  taken,  is  negative,  to 
add  it,  as  if  it  were  aflirmative,  to  9  ;  and  then  take  the 
re?t  of  the  figures  from  9,  as  before, 


HULE  OF  THREE  BY  LOGARITHMS.     275 


EXAMPLES. 

1.  Find  a  tbiuth  proportional  to  37.125,  14.768,  and 
135.279,  by  logarithms. 

Log.  of  37.125    ....  1.5696665 

Complement 8.4303335 

Log.  of  14.768     ....  1.1693217 

Log.  of  135.279        .     .     .  2.13^2304 

Ans.  53.81099       .     .     .  1.7308856 

2.  Find  a  fourth   proportional  to  .05764,  .7186,  and 
34721,  by  logarithms. 

Log.  of  .05764    ....  2.7607240 

Complement 11.2  392760 

Log.  of  .7186      ....  T.8564872 

Log.  of  .34721     .     .     .      ,  T.5405Q22 

Ans.  4.328681      .      .      .  0.636:3554 


3.  Find  a  third  proportional  to  12.796  and  3.24718,  by 
ms. 

Log.  of  12.796    ....      1,1070742 


logarithms 


Complement 8.8929258 

Log.  of  3.24718       .      .     .  0.5115064 

liOg.  of  3.24718       .     .     .  0.5115064 

Ans.   .824021G       .     .  T.91593S6 

4.   Find  the  interest   of  279/.  5s.  for  274  days,  at  4,;, 
per  cent,  per  annum,  by  logarithms. 


!^f6        INVOLUTION  BY  LOGARITHMS. 


Comp.  log.  of  100 
Comp.  loij.  of  366 
Log.  ot  279.25  . 
hovr.  of  274  .  . 
Lo^.  of  4.5 

Ans.  9.433296  . 


8.0000000 
7.4377071 
2.4459932 
2.4377.506 
0.6532125 

0.9746634 


5.  Find   a  fourth   proportional  to 
100.979,  by  lojrarilhms. 

6.  Find    a  tomth   proportional   to 
50.4567,  by  logaritJims. 

7.  Find  a   fourth    proportional   to 
.008967.  by  logarithms 


12.678,  14.065,  and 

Ans.  112.0263. 

1.9864,  .4678,   and 

Ans.   1 -.88262. 

.09658,  .2»958,   and 

Ans.  .02317234. 


8.  Fnid  a  mean  proportional  between  49S62I  and 
2..)587,  and  a  third  proportional  to  12.796  and  3.24718 
by  logarithms.  Ans.  17.55623  and  .8240216. 


INVOLUTION, 

OR  THE  RAISING  OF  POWERS  BY  LOGARITHMS. 

Take  out  the  logarithm  of  the  given  number  from  the 
tables,  and  multiply  it  by  the  index  of  the  proposed  pow- 
er; then  the  natural  number  answering  to  the  result,  will 
be  the  power  required. 

Observing,  if  the  index  of  the  logarithm  be  negative, 
that  this  part  of  the  product  will  be  negative  ;  but  as  what 
is  to  be  carried  from  the  decimal  part  will  be  affirmative, 
the  index  of  the  result  must  be  taken  accordingly. 

EXAMPLES. 


1.  Find  the  square  of  2.7568,  by  logarithms. 


INVOLUTION  BY  LOGARITHMS.        '2T 

Log.  of  2.7568       ,     .     .       0.4402477 


Square  7.599946  .     .    ".       0.8804^54 


i.  Find  the  cube  of  7.0851,  by  logarithms. 
Log.  of  7.0S51       ,     .     .      0.S5U3399 


3 


Cube  355.6475      ,     .     .       2,5510197 
3.  Find  the  fifth  power  of  .87451,  by  logarithms. 
Log.  of  .87451       .     .     .       T.94I7648 

5 

Fifth  power  .5114695      .      T.7088240 


Where  5  times  the  negative  index  —  1,   being  — 5,   and 
i-4  to  carry,  the  index  of  the  power  is  1. 

4,  Find  the  365th  power  of  1.0045,  by  logarithms. 
Log.  1.0045*     ....     0.0019499 

365 


97495 
116994 

58497 


Power  5.148888     .     .     .      0.7117135 


5.  Required  the  square  of  6.05987,  by  logarithms. 

Ans.  36.72203. 
0.  Required  the  cube  of  .176546,  by  logarithms. 

Ans.  .005502674, 

*  This  answer  5.148888,  though  found '(rirtly  according  to  the  general 
rule,  is  not  correct  in  the  last  four  figures  8888 ;  nor  can  the  answers  to  such 
■questions  relating  to  very  high  powers  be  generally  found  true  to  6  places  of 
figures  by  the  tables  of  Log.  commonly  used  ;  if  any  power  above  the  hun- 
dred thousandth  were  required,  not  one  figure  of  the  answer  here  given 
could  be  depended  on.  The  Log.  of  L0045  is  00194994108  true  to  eleven 
places,  which  multiplied  by  365  girc  s  .7117285  true  to  7  places,  and  the  cor- 
responding- number  true  to  7  places  is  5.149067.  See  Doctor  Adrain's  edi- 
tion of  Hut.  Math.  Vol.  1.  p.  169. 

B  b 


!78         EVOLUTION  BY  LOGARITHMS. 

7.  Required  the  4th  power  of  .076543,  by  logarithms. 

Ans.  .0000343259. 

8.  Required  the  5th  power  of  2.97643,  by  logarithms. 

Ans.  2.-j3.60:51 

9.  Required  the  6th  power  of  21.057»j,  by  logarithms. 

Ans.  fci71S7340, 

10.  Required  the  7th  power  of  1.09684,  by  logarithms. 

Ans.  1.909b64. 


EVOLUTION, 


OR  THE  EXTRACTION  OF  ROOTS,  BY  LOGARITHMS. 

Take  out  the  logarithm  of  the  given  number  from  the 
table,  and  divide  it  by  i',  for  the  square  root,  3  for  the 
cube  root,  &c.  and  the  natural  nuniber  answering  to  the 
result  will  be  the  root  required. 

But  if  it  be  a  compound  root,  or  one  that  consists  both 
of  a  root  and  a  power,  multiply  the  logarithm  of  the  given 
number  by  the  numerator  of  the  index,  and  divide  the  pro- 
duct by  the  denominator,  for  the  logarithm  of  the  root 
sought. 

Observing,  in  either  case,  when  the  index  of  the  loga- 
rithm is  negative,  and  cannot  be  divided  without  a  remain- 
der to  increase  it  by  such  a  number  as  will  render  it  ex- 
actly divisible  ;  and  then  carry  the  units  borrowed,  as  so 
many  tens,  to  the  first  figure  of  the  decimal  part,  and  di- 
vide the  whole  accordingly. 

KXAMPLES, 

L  Find  the  square  root  of  27.465,  by  logarithms. 
Log.  of  27.465  .     .      .     2j  1.4387796 


Root  5.2407     .....       .7193898 


EVOLUTION  BY  LOGARITHMS.  279 

2.   Find  the  cube  root  of  35.b4l5,  by  logarithms. 
Log.  of  35.9415    .      .      .     3). .5. 19560 

Root  3.29093 5173186 


Find  the  5th  root  of  7.0S25,  by  lo^rariihrns. 
Log.  of  7.U825       .      .      .      5)0.8501  »b6 


Root  1.47^235 1700373 

Find  the  365th  root  of  1,045,  by  logarithms. 
Log.  of  1.045    .      .      .      365)0.00 1 9499 

Root  1.000121      ....      0.u0n0>34 


5.  Find  the  vahie  of  (.001234)3  by  logarithms. 
Log.  of  .001234        .      .     .     3.0913152 


3)»^.1826S04 


Ans.  .00115047  ....     2.060876i\ 
Here,  the  divisor  3  being  contained  exactly  twice  in  the 
negative  index  — 6,  the  index   of  the  quotient,    to  be  put 
down,  will  be  — 2. 

3. 

Find  the  value  of  (.024554)2  by  logarithms. 
Log.  of  .024554       .     .     .     2.3901223 


2)5.1703669 


Ans.  .00384''54  ....     3.5851834. 
Here  2  not  being  contained  exactly  in  -  5,  1  is  added  to 
ft,  which  gives  —3  for  the  quotient ;  and  the  1  that  is  bor» 


280  QUESTIONS  IN  LOGARITHMS. 

rowed  being  carried  to  the  next  figure  makes    11,  whicft.- 
divided  by  2,  gives  .58,  &c. 

7.  Required  ihe  square  root  of  365.5674,  by  logarithms 

Ans.    19.11981, 

8.  Required  the  cube  root  of  2.987635,  by  logarithms. 

Ans.   1.440265, 

9.  Required  the  4th  root  of  .967845,  by  logarithms. 

Ans.    .9918^^2-1. 

10.  Required  the  7th  root  of  .(98674,  by  logarithms. 

Ans.   .7183146. 

11.  Required  the  value  of  (-r^)^,  by  logarithms. 

Ans.  .1-16895.- 
-——)",  by  logarithms. 

Ans.  .1937115^ 

MISCELLANEOUS  EXAMPLES  IN  LOGARITHBIS. 

\,  Required  the  square  root  of  — - .  by  logarithms. 

Ans.  .1275153. 

2.  Required  the  cube  root  of; — ,  by  logarithms. 

Ans.   .682784?. 
3^.  Required  the  .07  power  of  .OOof  3,  by  logarithms, 

Ans.   .6958821 

4.  Required  the  value  of  -^ — ^— ^    »  by  logarithms. 

Ans.   .04279825. 

15  7 

5.  Required  the  value  of -.v/-X.012  3/— ,    by    loga- 

7      8  11 

lithms.  Ans.   .001165713. 

C.  Required  the  vaU.e  of  ^l^'^lj^i^i  by  loga- 

rithm.?,  Ans,  .300915863S. 


MISCELLANEOUS  QUESTIONS.  281 

7.  Kequired   the   value  of 1  ^- ---^- l>by 

logarithms.  Ans.  49.38712. 

MISCELLAiNEOUS  QUESTIONS. 


1.  A  person  being  asked  what  o'clock  it  was,  replied 
that  it  was  between  ei^iht  and  nine,  and  that  the  hour  and 
minute  hands  were  exactly  together ;   what  was  the  time  ? 

Ans.  8h.  43  min.  38 -fy  sec. 

2.  A  certain  number,  consisting  of  two  places  of  figures, 
u  equal  to  the  difference  of  the  squares  of  its  digits,  and 
if  36  be  added  to  it  the  digits  will  be  inverted  ;  what  is  the 
number  1  Ans.  48. 

3.  What  two  numbers  are  those,  whose  difference,  sum, 
and  product,  are  to  each  other  as  the  numbers  2,  3,  and  5, 
respectively  ?  Ans.  2  and  10. 

4.  A  person,  in  a  party  at  cards,  betted  three  shillings 
to  two  upon  every  deal,  and  after  twenty  deals  found  he 
had  gained  five  shdlings  ;  how  many  deals  did  he  win  ? 

Ans.  13. 

5.  A  person  wishing  to  enclose  a  piece  ©f  ground  with 
palisades,  found,  if  he  set  them  a  foot  asunder,  that  he 
should  have  too  few  by  150,  but  if  he  set  them  a  yard 
asunder  he  should  have  too  many  by  70 ;  how  many  had 
he?  '  Ans.  180. 

6.  A  cistern  will  be  filled  hy  two  cocks,  a  and  b,  run- 
ning together,  in  twelve  hours,  and  by  the  cock  a  alone  in 
twenty  hours  ;  in  what  time  will  it  be  filled  by  the  CGck  b 
alone  1  Ans.  30  hours. 

7.  If  three  agents,  a,  b,  c,  can  produce  the  effects  a,  b, 
<:,  in  the  times  e,f,  g,  respectively ;  in  what  time  would 
<.hey  jointly  produce  the  effect  d. 


Ans.  d 
Bb2 


*(H+i) 


182         MISCELLANEOUS  QUESTIONS. 

8.  What  number  is  that,  which  being  severally  added 
to  3,  19,  and  51,  shall  make  the  results  in  geometrical  pro* 
gression  ?  Ans.  13. 

9.  It  is  required  to  find  two  geometrical  mean  propor- 
tionals between  three  and  24,  and  four  geometrical  means 
between  3  and  96. 

Ans.  6  and  12 ;  and  6,  12,  24,  and  48. 

10.  It  is  required  to  fiiid  six  numbt.-rs  in  geometrical 
progression  such,  that  their  sum  shall  be  315,  and  the  sum 
of  the  two  extremes  165. 

Ans.  5,  10,  20,  40,  80,  and  160. 

11.  The  sum  of  two  numbers  is  a,  and  the  sum  of  their 
reciprocals  is  b,  required  the  numbers. 


A„s.|±l^J|(ai-4)§ 


12.  After  a  certain  number  of  men  had  been  employed 
on  a  piece  of  work  for  24  days,  and  had  half  finished  it, 
16  men  more  were  set  on,  by  which  the  remaining  half 
was  completed  in  16  days;  how  many  men  were  emjjloy- 
ed  at  first;  and  what  was  the  whole  expense,  at  I5.  6c?.  a 
day  per  man?  Ans.  32  the  number  of  men  ;  and  the 

whole  expense  115/.  4?. 

13.  It  is  required  to  find  two  numbers  such,  that  if  the 
squares  of  the  first  be  added  to  the  second,  the  sum  shall 
be  62,  and  if  the  square  of  the  second  be  added  to  the 
first,  it  shall  be  176.  Ans.  7  and  13. 

14.  The  fore  wheel  of  a  carriage  makes  six  revolutions 
more  than  the  hind  wheel,  in  going  120  yards  ;  but  if  the 
circumference  of  each  wheel  was  increased  by  three  feet, 
it  would  make  only  four  revolutions  more  than  the  hind 
wheel  in  the  same  space  ;  what  is  the  circumference  of 
each  wheel  ?  Ans.  12  and  15  feet. 

15.  It  is  required  to  divide   a  given  number  a  into  two 

such  parts,  x  and  y,  that  the  sum  of  mx  and  vy  shall  be 

equal  to  some  other  given  number  b. 

.  ba — n       .        am — b 

Ans.  x= and  y= . 

m — n  m — n 

16.  Out  of  a  pipe  of  wine,  containing  84  gallons,  10 
gallons  were  drawn  oflT,  and  the  vessel  replenished  with 


'  MISCELLANEOUS  QUESTIONS.          283 

10  gallons  of  water;  after  which,  10  gallons  of  the  mix- 
ture were  again  drawn  off,  and  then  10  gallons  more  of  wa- 
ter poured  in ;  and  so  on  for  a  third  and  fourth  time ; 
which  being  done,  it  is  required  to  find  how  much  pure 
wine  remained  in  the  vessel,  supposing  the  two  fluids  to 
have  been  thoroughly  mixed  each  time  ?  Ans.  484  gallons. 

17.  A  sum  of  money  is  to  be  divided  equally  among  a 
certain  number  of  persons  ;  now  if  there  had  been  3  clai- 
mants less,  each  would  have  had  150/.  more,  and  if  there 
had  been  6  more,  each  would  have  had  15()Z.  less  ;  requir- 
ed the  number  of  persons,  and  the  sum  divided. 

Ans.  9  persons  ;    sum  2700/. 

18.  From  each  of  16  pieces  of  gold,  a  person  filed  the 
worth  of  half  a  crown,  and  then  offered  them  in  payment 
for  their  original  value,  but  the  fraud  being  detected,  and 
the  pieces  weighed,  they  were  found  to  be  worth,  in  the 
whole,  no  more  than  eight  guineas  ;  what  was  the  original 
value  of  each  piece?  Ans.  135. 

19.  A  composition  of  tin  and  copper,  containing  100 
vubic  inches,  was  found  to  weigh  505  ounces  ;  how  many 
ounces  of  each  did  it  contain,  supposing  the  weight  of  a 
cubic  inch  of  copper  to  be  5|  ounces,  and  that  of  a  cubic 
inch  of  tin  4^  ounces. 

Ans.  420  oz.  of  copper,  and  85  oz.  of  tin, 

20.  A  privateer,  running  at  the  rate  of  10  miles  an  hour, 
discovers  a  vessel  IB  miles  ahead  of  her,  making  way  at 
the  rate  of  8  miles  an  hour  ;  how  many  miles  will  the  lat- 
ter run  before  she  is  overtaken.  Ans.  72  miles. 

21.  In  how  many  different  ways  is  it  possible  to  pay 
100/.  with  seven  shilling  pieces,  and  dollars  of  4s.  6d. 
«ach  ?  Ans.  31  different  ways. 

22.  Given  the  sum  of  2  numbers  =2,  and  the  sum  of 
their  ninth  powers  =32,  to  find  the  numbers  by  a  quad- 
ratic equation.  Ans.  1  ±1^(6^34  —  38). 

23.  Given  ?/'-x?/=666,  and  xM-'^!/  — 406 ;  to  find  x 
and  y.  Ans.  a:=7,  and  ^=9. 

24.  The  arithmetical  mean  of  two  numbers  exceeds  the 
geometrical  mean  by  13,  and  the  geometrical  mean  ex- 
ceeds the  harmonical  mean  by  12  ;  wHat  are  the  numbers ; 

Ans.  234  jin<^  104. 


284  MISCELLANEOUS  QUESTIONS. 

25.  Given  x^3/-fi/^a;=3,  and  a:y+2/V=7,  to  find  the 
values  of  x  and  y. 

Ans.  x-=i(v/o-rl),  ^/=i(v/.^-l). 

26.  Given  x-\-y-^z=2S,  cry-}  xz-\-yz  =  l67,  and  3yz=^ 
385,  to  find  x,  y,  and  z.  Ans.  a;=.^,  3/==7,  z  =  l  I. 

27.  To  find  fin-  numbers,  t,  i/,  ^,  and  a',  having  the 
product  of  every  >  ree  of  them  given;  viz.  j-//2=^:^ol, 
a:^te'="420,  ?/ea?=l    40,  and  t2'm==6<'0. 

Ans.  x—3. 3/^7,  2=1 1 ,  and  Zi)  —  20. 

28.  Given  a:-|-2/2^38-i,  2/4-a:e  =  237,  and  ^-^J?/-19^, 
to  findjhe  values  of  x,  y,  and  2-. 

Ans.  x=  10,  ?,'=17,  and  ^  — 22. 

29.  Given  2^-1-./]/  =  108.  3/  ^3/2=69,  and  x''-[-xz=bm, 
to  find  the  vakies  of  x,  y,  and  ^. 

Ans.  x=9, 2/=3,  and  z==20. 

30.  Given  3:^+x^  +  ?/-=o,  and  x^+a;^?/^+?/^=  1 1,  to  find 
the  values  of  x  and  ?/  by  a  quadratic. 

Ans.  X  =^y  I  O-f-  ^  5,  t/=-  yj  10— -^/5. 

31.  Given  the  equation  x^ — Sx^-f-l  3x^ — 12x  =  5,  to  find 

3      1 

the  value  of  j  by  a  quadratic.  Ans.  -±-,y  13. 

32.  It  is  required  to  find  by  what  part  of  the  population 
a  people  must  increase  annually,  so  that  they  may  be  dou- 
ble at  the  end  of  every  century. 

Ans.  By  144th  part  nearly, 

33.  Required  the  least  numi.er  of  weights',  and  the 
weight  of  each,  that  will  wei^h  any  number  of  pounds 
from  one  pound  to  a  hundred  weight. 

Ans.  1,3,  9,  27,  8!. 

31.  It  is  required  to  find  foui  whole  numbers  such,  that 
the  square  of  the  greatest  may  be  equal  to  the  sum  of  the 
squares  of  the  other  three.  Ans.  3,  4,  \^,  and  13. 

35.  It  is  required  to  find  the  least  number,  which  being 
divided  by  6,  5,  4,  3,  and  2,  shall  leave  the  remainders  5, 
4,  3,  2,  and  1,  respectively.  Ans.  5D. 

3G.  Given  the  cycle  of  the  sum  18,  the  golden  number 
v*>,  and  the  Rpman  indiction  10,  to  find  the  year. 

Ans.  1717. 


MISCELLANEOUS  QUESTIONS.         285 

37.  Given  2o6r — 87v=l,  to  find  the  least  possible  values 
of  a;  and  y  in  whole  numbers.         Ans.  a-=52,  and  ^=153. 

38.  It  is  required  to  find  two  diffprent  isosceles  trian- 
gles such,  that  their  perim^^ters  and  areas  shall  be  both 
expressed  by  the  same  numbers. 

Ans.  Sides  of  the  one  ^29,  29,  .0  ;  and  of  the  other  37,  37, 24. 

39.  It  is  required  to  find  the  sides  of  three  right  angled 
triangles,  in  whole  numbers,  such,  that  their  areas  shall 
be  all  equal  to  each  other. 

Ans.  58,  40,  42  :   74,  24,  70  ;  113,  15,  112. 

40.  Given  ic^  =  1.2655,  to  find  a  near  approximate  va- 
lue of  X.  Anir^.  O.82013. 

41.  Given  a:2/=5O00,  and  2/^^=3000,  to  find  the  values 
of  a:  and  y.  Ans.  a:=4.6'->l  445,  and  y=  ">.5iO\'62. 

42.  Given  x^'-^-yy  =285,  and  ^^  — t2/=  I4,  to  find  the  va- 
lues of  X  and  y.  Ans.  .r—4.Ul 66 -'8,  and  </  — 2.8257  16. 

43.  To  find  two  whole  numbers  such,  that  if  unity  be 
added  to  each  of  them,  and  also  to  their  halves,  the  sums, 
in  both  cases,  shall  be  squares.  Ans.  48  and  1680. 

44.  Required  the  two  least  nonquadrate  numbers  x  and 
y  such,  that  x'^-{-y^  and  a:^4?/' shall  be  both  squares. 

Ans.  a:-=j64  and  2/=273. 

45.  It  is  required  to  find  two  whole  numbers  such,  that 
their  sum  shall  be  a  cube,  and  their  product  and  quotient 
squares.  Ans.  2o  and  lOw,  or  100  and  900,  &c. 

46.  It  is  required  to  find  three  biquadrate  numbers  such, 
that  their  sum  shall  be  a  square.         Ans.  \2\  1  o^,  and  20\ 

47.  It  is  requiif'd  to  find  three  numbers  in  continued 
geometrical  progression  such,  that  their  three  differences 
shall  be  all  squares.  Ans.  567,  100§,  and  1792. 

413.  It  is  required  to  find  three  whole  numbers  such,  that 
the  sum  or  difference  of  any  two  of  them  shall  be  square 
numbers.  Ans.  434:57,  420968,  and  150568. 

49.  It  is  required  to  find  two  whole  numbers  such,  that 
their  sum  shall  be  a  square,  and  the  sum  of  their  squares 
a  biquadrate.      Ans  4565486027761  and  1061652293520. 

50.  It  is  required  to  find  four  whole  numbers  ouch,  that 
the  difterence  of  every  two  of  them  shall  be  a  square  nam- 
here         Ans.  1873432,  2288168,  2399057,  and  C560657. 


286  MISCELLANEOUS  QUESTIONS. 

5J.  It  is   required   to  find   the  sum  of  the  series --{-- 

3       4  3 

+r^+^+&c.  continued  to  infinity.  Ans.-. 

52.  It  is  required  to  find  tlie  sum  of  the  infinite  series 
3      9       27       81        243    .  ^3 

4-i^+6i~256  +  Ki24  ^"^  ^""''l' 

63.  Required  the  sum  of  the   series  54-6  +  74-84-94" 

&c.  continued  to  n  terms.  Ans.  -(n+9). 

54.  It  is  required  to  find  how  many  figures  it  would  take 
to  express  the  "^5th  term  of  the  series  2' -f  2--f2^-f23  + 
2^'  &c.  Ans.  50  044H  figures. 

55.  It  is  required  to  find  the  sum  of  100  terms  of  the 
series  (1  X2)  +  (3X4)  +  (5X6)+(7Xi  )4-(9XlO)  &c. 

Ans.  l:-543300. 

56.  Required  the  sum  of  i24-q2_j. -.2^42^  ,2  &c.  .  .  . 

.  4*60  which   gives  the  number  of  shot  in   a  square  pile, 
the  side  of  which  is  50.  Ans.  42925. 

57.  Required  the  sum  of  25  terms  of  the  series  35436 
X2-f37X3-f  38X4  +  39X5  &c.  which  gives  the  number 
of  shot  in  a  complete  oblong  pile,  c(i«sisting  of  25  tiers, 
the  number  of  shot  in  the  uppermost  row  being  35. 

Ans.  16575. 


28" 


APPENDIX. 


OF  THE  APPLICATION  OF  ALGEBRA  TO 
GEOMETRY. 

In  the  preceding  part  of  the  present  performance,  I  have 
considered  Algebra  as  an  independent  science,  and  confin= 
ed  myself  chiefly  to  the  treating  on  such  of  its  most  useful 
rules  and  operations  as  could  be  brought  within  a  moderate 
compass  ;  but  as  the  numerous  apphcations,  of  which  it  is 
susceptible,  ought  not  to  be  wholly  overlooked,  I  shall  here 
shovy,  in  compliance  with  the  wishes  of  many  respectable 
teachers,  its  use  in  the  resolution  of  geometrical  prob- 
lems ;  referring  the  reader  to  my  larger  work  on  this  sub- 
ject, for  what  relates  more  immediately  to  the  general 
doctrine  of  curves.* 

For  this  purpose  it  may  be  observed,  that  when  any 
proposition  of  the  kind  here  mentioned  is  required  to  be 
resolved  algebraically,  it  will  be  necessary,  in  the  first 
place,  to  draw  a  figure  that  shall  represent  the  several 
parts,  or  conditions,  of  the  problem  under  consideration, 
and  to  regard  it  as  the  true  one.' 

Then,   having   properly   considered  the   nature   of  the 


*  Tiie  learner,  before  he  can  obtain  a  competent  knowledge  of  the  method 
of  application  above  mentioned,  must  first  make  himself  master  of  the  prin- 
cipal propositions  of  Euclid,  or  of  those  contained  in  my  Elements  of  Geo- 
metry;  in  which  nork  he  will  find  all  the  essential  principles  of  the  science 
comprised  within  a  much  shorter  compass  than  in  the  former. 

And  in  such  cases  where  it  may  be  reijuisite  to  extend  this  mode  of  appli- 
cation to  trigonometry,  mechanics,  or  any  other  branch  of  mathematics,  a  pre- 
vious knowledge  of  the  nature  and  principles  of  these  subjects  will  be  equal- 
'y  necessary. 


288  APPLICATION  OF 

question,  the  figure  so  formed,  must,  if  necessary,  be  stiil 
farther  prepared  for  solution,  by  producing,  or  drawing, 
such  lines  in  it  as  may  appear,  by  their  connexion  or  rela- 
tions 10  each  other,  to  be  most  conducive  to  the  end  pro- 
posed. 

This  being  done,  let  the  unknown  line,  or  lines,  which 
it  is  judged  will  be  the  easiest  to  find,  together  with  those 
that  are  known,  be  denoted  by  the  common  algebraical 
symbols,  or  letters ;  then,  by  means  of  the  proper  geome- 
trical theorems,  make  out  as  many  independent  equations 
as  there  are  unknown  quantities  employed  ;  and  the  reso- 
lution of  these,  in  the  usual  manner,  will  give  the  solution 
of  the  problem. 

But  as  no  general  rules  can  be  laid  down  for  drawing 
the  lines  here  mentioned,  and  selecting  the  properest  quan- 
tities to  substitute  for,  so  as  to  bring  out  the  most  simple 
conclusions,  the  best  means  of  obtaining  experience  in 
these  matters  will  be  to  try  the  solution  of  the  same  prob- 
lem in  different  ways  ;  and  then  to  apply  that  which  suc- 
ceeds the  best  to  other  cases  of  the  same  kind,  when  they 
afterwards  occur. 

The  following  directions,  however,  which  are  extracted, 
with  some  alterations,  from  Newton's  Universal  Arithfnetic, 
and  Simpson's  Algebra  and  Select  Exercises,  will  often  be 
tound  of  considerable  use  to  the  learner,  by  showing  him 
how  to  proceed  in  many  cases  of  this  kind,  where  he 
would  otherwise  be  left  to  his  own  judgment. 

1st.  In  preparing  the  figure  in  the  manner  above  men- 
tioned, by  producing  or  drawing  certain  lines,  let  them  be 
either  parallel  or  perpendicular  to  some  other  lines  in  it, 
or  be  so  drawn  as  to  form  similar  triangles  ;  and,  if  an  an- 
gle be  given,  let  the  perpendicular  be  drawn  opposite  to 
it,  and  so  as  to  fall,  if  possible,  from  onu  end  of  a  given 
line, 

2d.  In  selecting  the  proper  quantities  to  substitute  for, 
let  those  be  chosen,  whether  required  or  not,  that  are 
nearest  to  the  known  or  given  parts  of  the  figure,  and  by 
means  of  which  the  next  adjacent  parts  may  be  obtained 
by  addition  or  subtraction  only,  without  using  surds. 


ALGEBRA  TO  GEOMETRY. 


289 


Bd.  When  in  any  problem,  there  are  two  lines,  or  quan- 
tities, alike  related  to  other  uarts  of  the  fi'iure,  or  problem, 
the  best  way  is  not  to  make  use  of  either  of  them  sepa- 
rately, but  to  substitute  for  their  sum,  difference,  or  rect- 
angle, or  the  sum  of  their  r.lternate  quofi.^nts  ;  or  for  some 
other  line  or  lines  in  the  figure,  to  which  they  have  both 
the  same  relation. 

4th.  When  the  area,  or  the  permieter,  of  a  figure  is 
given,  or  such  parts  of  it  as  have  or*ly  a  remote  relation  to 
the  parts  that  are  to  be  found,  it  will  sometimes  be  of  use 
to  assume  another  fi^jure  similar  to  ihe  proposed  one,  that 
shall  have  one  of  its  sides  equal  to  unity,  or  to  some  other 
known  quantity  ;  as  the  other  parts  of  the  figure,  in  such 
cases,  may  then  be  determined  l»y  the  known  proportions 
of  their  like  sides,  or  parts,  and  tiience  the  resulting  equa- 
tion required. 

These  being  the  most  general  observations  that  have 
hitherto  been  collected  upon  this  subject,  I  shall  now  pro- 
ceed to  elucidate  them  by  proper  examples  ;  leaving  such 
farther  remarks  as  may  arise  out  of  the  mode  of  proceed- 
ing here  used,  to  be  applied  by  the  learner,  as  occasion 
requires,  to  the  solutions  of  the  miscellaneous  problems 
given  at  the  end  of  the  present  article. 

PROBLEM  r. 

The  base,  and  the  sum  of  the  hypothenuse  and  perpen^ 
dicular  of  a  right  angled  triangle  being  given,  it  is  requir^ 
ud  to  determine  the  triangle. 


A 


Let  ABC,  right  angled  at  c,  be  the  proposed  triangle  ;  an<i 
\mi  BC—b,  and  AC=a:. 


290 


APPLICATION  OF 


Then,  if  the  sum  of  ab  and  ac  be  represented  by  s,  the 
h^^pothenuse  ab  will  be  cxpres.-^ed  by  s~x. 

Bill,  by  the  well  known  property  of  right  angled  trian- 
gles (Euc.  I.  47.) 

AC-4-BC^=AB^,  or 

Whence,  onaitting  a^,  which  is  common  to  both  sides  of 
the  equation,  and  transposing  the  other  terms,  we  shall 
have  2sx=s^  —  b^,  or 

^  2s     '  *  *  *  ' 

which  is  the  value  of  the  perpendicular  ac  ;  where  a  and 
b  may  be  any  numbers  whatever,  provided  s  be  greater 
than  6. 

In  like  manner,  if  the  base  and  the  difference  between 
the  hypothenuse  and  perpendicular  be  jiiven,  we  shall 
have,  by  putting  x  for  the  perpendicular  and  ti+x  for  the 
hypothcDuse, 

a>2+2tix-i-£/==62_|_^2^  or 


2cl 
Where  the  base   {l)  and  the  given  difterence  (J)   may 
be  any  numbers  as  before,  provided  b  be  greater  than  d. 

PROBLEM  II. 

The  difference  between  the  diagonal  of  a   square  ano 
one  of  its  sides  being  given,  to  determme  the  square- 


*  The  edition  of  Euclid,  referred  to  in  this  and  all  the  following  problems. 
13  that  of  Dr.  Simson,  London,  1801 ;  which  may  also  be  used  in  the  geome- 
trical construction  of  these  problems,  should  the  student  be  inclined  to  exer- 
ciS«  histalenlS  upon  this  elegant,  but  more  difficult  branch  of  the  fufcjct!. 


ALGEBRA  TO  GEOMETRY.  291 

Let  AC  be  the  proposed  square,  and  put  the  side  bc,  or 
■  '),  ~x. 

Then,  if  the  difference  of  bd  and  bc  be  put  =d,' the 
hypoihenuse  bd  will  be  =x-\-d. 

But  Since,  as  in  the  former  problem,  bc^+cd^  or  2bc- 
=  BD",  we  shall  liave 

Which  equation  being  resolved  according  to  the  rule 
laid  down  for  quaHratics,  in  the  preceding  part  of  the 
work,  gives 

Which  is  the  value  of  the  side  sc,  as  was  required. 

PROBLEM  111. 

The  diagonal  of  a  rectangle  abcd,  and  the  perimeter, 
>r  Sum  of  all  its  four  sides,  being  given,  to  find  the  sides. 


Let  the  diaijonal  Ac=rf,  half  the  perimeter   AB-|-BC=a, 
and  the  base  B(j=x  ;  then  will  the  altitude  AB=a — x. 

And   since   as  in  the   former  problem,    ab^-{-bc^=ac^, 
we  shall  have 

a^  —  2ax-\rx^-k-x'^=d^,  or 

Which  last  equation,  being  resolved,  as  in  the  former 
instance,  gives 

x=^a±^^{2d^-a^). 
Where  a  must  be  taken  greater  thari  d  and  less  than  d^2, 


\92 


APPLICATION  Ot 

PROBLEM  IV. 


The  base  and  perpendirular  of  any  plane  triangle 
being  given,  to  hnd  the  side  of  its  inscribed  square. 


B      F    P    a     c 

Let  FG  be  the  inscribed  square  ;  and  put  bc  — 6,  ad=/?; 
and  the  side  of  the  .'(juare  eh  or  ef=^. 

Then,  because  the  triangles  abc,  aeh,  are  similar,  (Euc. 
VI,  4, J  we  shall  have 

ad  ;  Bc  ',',  Ai  :  EH,  or 
f  ■  b  ::  (p-oc)  :  X. 
Whence,  tskinj?  the  products   of  the  means  and   ex- 
tremes, there  will  arise 

px^bp^bx. 

Which  by  transposition  and  division,  gives 

bp 

Where  b  and  p  may  be  any  numbers  whatever,  either 
whole  or  fractional. 

PROBLEM  V. 

Havinfr  the  lengths  of  three  perpendiculars,  ef,  eg,  eh. 
drawn  from  a  certain  point  r,  within  an  equilateral  triangle 
ABC,  to  its  three  sides,  to  determine  the  sides. 


ALGEBRA  TO  GEOMETRY. 


293 


Draw  the  perpendicular  ad,  and  having  joined  ea,  ee, 
and  EG,  put  EF— a,  eg^=6,   eh=:c,  and  bd  (which  is  ^bc) 

Then,  since  ab,  eg,  or  ca,  are  each  =2r,  we  shall  have, 
by  Euc.  1,  47, 

AD=/(AB2-BD2)=y(4x2-x')=v^3x2=Xv/3. 

And  because  the  area  of  any  plane  triangle  is  equal  to 
half  the  rectangle  of  its  base  and  perpendicular,  it  fol- 
lows,  that 

A  ABC=^Bc  X  .\D=;r  Xa;y/3=a?-^3, 

ABEC=^BCXbF=xXa         =^aXy 

AAR:c=iAc  Xkg3=xX6        =ssbxj 

AAEB=iABXEH=a;XC  =cx. 

But  the  last  three  triangles  beg,  aeg,  aeb,  are,  together, 
equal  to  the  whole  triangle  abc  ;  whence 
.r^^  3 -■=  ax -|- 6  X -{- ex , 
And  consequently,  if  each  side  of  this   equation  be  di- 
vided by  X,  we  shall  have 

x^3=a-{-b-\-c,  or 

Which  is,  therefore,  half  the  length  of  either  of  the 
three  equal  sides  of  the  triangle. 

Cor.  Since,  from  what  is  above  shown,  ad  is  =a;^3, 
it  follows,  that  the  sum  of  all  the  perpendiculars,  drawn 
from  any  point  in  an  equilateral  triangle  to  each  of  its  sides; 
is  equal  to  the  whole  perpendicular  of  the  triangle. 

PROBLEM  VI. 

Through  a  given  point  p,  in  a  given  eircle  agbd,  to  draw 
a,  chord  cd,  of  a  given  length. 

A 


^94  APPLICATION  OF 

Draw  the  diameter  apb  ;  and  put  cD=a,  ap=6,  pb=c. 
and  iP^x  ;  then  vviil  rD=^a--T. 

But,  by  the  property  of  the  circle  (Euc.  iti,  36,)  cp  X 
rD=APXPB;  whence 

x{a — x)=bcy  or 
x"*^— ax=  —  6c. 
Which  equation,  being  resolved  in  the  usual  way,  give? 

x  =  ia±  ^/{la'—bc)  ; 
V\  here  x  has  two  values,  both  of  which  are  positive. 

PROBLEiVI  VII. 

Through  a  given  point  p,  without  a  given  circle  abdc,  to 
draw  a  right  line  so  that  the  part  en,  intercepted  by  the 
circumference,  shall  be  of  a  given  length. 


Draw  PAB  through  the  centre  o  ;  and  put  cB  =  a,  -pk  —  b, 
pB=c.  and  Pc  —  o:  ;  then  will  pD~.v-f-«- 

But,  by  the  property  of  the  circle,    (Eic.  in,  36,  cor.,^ 
pcXpd=paXpb;  whence 

a(x-4-o)=6c,  or 
x^-\-ax~bc. 
Which  equation   being  resolved,  as  in  the  former  prob 
lem,  gives 

x  =  ~- {a± ^/ {\a--\-hc) ; 
Where  one   value  of  x  is  poshive  and  the  other  nega- 
tive.* 


*  The  two  last  problems,  with  a  (ew  slight  alterations,  may  be  readilj  ci< 
ployed  for  finding  the  roots  of  quadratic  tquaticn*  by  construction  ;  but  thii, 
as  well  as  the  methods  frequently  given  for  coi  s'ru'.ting  cubic  and  some  o.' 
the  higher  order  of  equations,  is  a  matter  of  little  importance  in  the  present 
state  of  mathematical  science  ;  anal}sis,  in  these  cases,  being  generally 
thought  a  more  commodious  instrument  than  geometry. 


ALGEBRA  TO  GEOMETRY. 


29a 


Pf  OBLEM  Vlll. 


The  base  bc,  of  any  plane  triani^le  abc,  the  sum  of  the 
sides  AB,  AC,  and  the  hue  ao,  drawn  fronn  the  vertex  to  the 
middle  of  the  base,  bemg  given,  to  determine  the  triangle. 


Put  BD  or  DC=a,  AD  =  6,  AB-|-AC=5,  and  AB=a;  ;  theft 

will  AC=S  — a\ 

But,  by  my  Geometry,  B^  ii.,  Prop.  19,  AB--f  ac^=2bd- 
-f-2AD  ;  whence 

x^+{s^xY—'2a--{-2h^,  or 


x^ — sx=a^-{-b~  — 


i«2 


Which   last  equation,  beiui^  resolved  as  in    the  former 
instances,  gives 

for  the  values  of  the  two  sides  ab  and  ac  of  the  triangle  ; 
taking  the  sign  +  for  one  of  them,  and  ~  for  the  other, 
and  observing  that  a^-^-b"  must  be  greater  than  {s^. 


PROBLEM  IX. 


The  two  sides  ab,  ac,  and  the  line  ad,  bisecting  the  ver- 
tical angle  of  any  plane  triangle,  abc,  being  given,  to 
find  the  base  BC. 


296 


APPLICATION  OF 


Put  AB=a,  Ac=6,  AD=c,  and  BC=a;  ;  then,  by  Euc.  Vi 
3,  we  shall  have 

AB(a)  :  Ac(6)  ::  BD  :  dc. 

And,  consequently,  by  the  composition  of  ratios  (Euc.  v. 
18,) 

a-\-b  :  a  \\  X  :  bd^ 


a^b  :  b 


and 

:    X   ;   DC: 


'a+6' 
bx 


But,  by  Euc.  vi,   13,  bdXdc-{-ad2=abXac  ;  where- 
fore, also, 

—-Jrc^=.ab,  or 

a6x2=(a4-6)2X(a6-c2). 
From  which  last  equation  we  have 
,     ,  ,,     «6  — c2 

Which  is  the  value  of  the  base  bc,  as  required. 

PROBLEM  X. 


Having  given  the  lengths  of  two  lines  ad,  bp:,  drawn 
from  the  acute  angles  of  a  right  angled  triangle  abc,  to  the 
middle  of  the  opposite  sides,  it  is  required  to  determine 
the  triangle. 

A 


B  I)  C 

Put  AD=a,  Be=6,  CD  or  icB=a:,  and  ce  or  icA=2/  ; 
then,  since  (Euc.  i,  47)  cd'^+ca2=ad2,  and  ce^+cb^= 
BE^  we  shall  have 


ALGEBRA  TO  GEOMETRY. 


297 


Whence,  taking  the  second  of  these  equations  from  fouY 
times  the  first,  there  will  aii>e 

1   v  — 4u^  — 6^,  or 

And,  in  like  manner,   iakinji  the  first  of  the  same  equa- 
tions from  four  times  the  serond,  there  will  arise 
;5i-=r4c/^   _a^j  or 

Which  values  of  ar  and  y  are  half  the  lengths  of  the  base., 
and  perpendicular  of  ttte  tnanjjie  ;  observing  that  k  must 
be  less  than  ^la,  and  greater  than  J«. 

PROBLEM  XI. 

Having  given  the  ratio  nf  the  two  sides  of  a  plane  tri^ 
angle  ab'^,  and  the  segments  of  the  base,  made  by  a  per- 
pendicular falhng  from  the  vertical  angle,  to  determine  the 
triangle. 


Put  BD=a,  Dc~6,  AB=rr,  AC=y,  and  the  ratio  of  the 
sides  as  m  to  «. 

Then,  since  by  the  question,  ab  :  ac  ::  m:  h,  and  by 
B.  11,  Prop.   iO*,  of  fny  Elements  of  Geometry^  ad^ — AC^= 


X  :  y  ['.  vi:  n,  and 

But,  by   the  first  of  these   expressions,  nx^=my,  or  ^= 

—  ;  whence,   if  this   be  substituted  for  y  in  the  second, 

m 

there  will  arise 


298 


APPLICATION  OF 


n2 
x"^ -a7^=a-~6-,  or 

Aud,  consequently,  by  division  and  extracting  the  square 
root,  we  shall  have 


x=m^ 


and 


62 


which  are  the  values  of  the  two  sides  ab,  ac,  of  the  trian- 
gle,  as  was  required. 

PROBLEM  Xll. 

Given  the  hypnthennse  of  a  right  angled  triangle  abc. 
and  the  sides  of  it.s  inscribed  square  do,  to  find  the  other 
two  sides  of  the  triangle. 


Put  AB=^,  Ds,  or  DF=?,  Ac=^j,  and  cb=^;  then,  by 
similar  triangles,  we  shall  have 

ac'.t)  :  CB  (//)  ::  af{x  —  s)  :  fb{s). 
And,   consequently,    by    multiplying  the  means  and  ex- 
tremesj 

xy     sy  =  sxj  or 

xy^s{x+y), (1) 

But  since,  by  Euc.  i,  47,   ac^+cb-^^ab^,  we    shall  like- 
wise have 

a:2  4-t/-=/i- (2) 

Whence,  if  twice  equation  (1)  be  added  to  equation  (2), 
there  will  arise 

x^'\'2xy-[-y^=h'^2s(x-\-y),  or 
{x+yY^2s{x-{-y)=h\ 


ALGEBRA  TO  GEOMETRY. 


199 


Which  question,  being  resolved  after  the  manner  of  a 
4uadratic,  gives 

x'-f  y=s±y'(/i^4-«^),  or 
y=s~x±  ^{k^'+s'') 
Hence,  if  this  value  be  substituted  for  y  in  equation  (I), 
there  will  arise 

x{s-!x:±y{h^-\-s^)\=sSs±  v/(/i'+s')|,  or 

And,  consequently,  by   resolving  this  last  equation,  we 
■shall  have 

and 

Which  are  the   values  of  the  perpendicular  ac  and  base 
r?c,  as  was  required. 


PROBLEM  xm. 


Having  given  the  perimeter  of  a  right  angled  triangle 
ABC,  and  the  perpendicular  cd,  falling  frcun  the  right  angle 
on  the  hypothenuse,  to  determine  the  triangle. 


Putp=  perimeter,  cD~a,  ac=.t,  and  Bc=y  ;  then  ab- 

But,  by  right  angled  triangles   (Euc.  i,  47)  ac^-j-bc^^— 
KB^ ;  whence 

Or,  by  transposing  the  terms  and  dividing  by  2, 
p{x'{-y)^^p^=xy (1). 


^0  APPLICATION  OF 

And  since,  by  similar  triangles,  ab  :  bc  ;•.  Ac  :  cv>,  we 
shall  also  have,  by   multiplM/,g  the  means  and  extremes 

AB  XcD     EC  Xac,  or 
^p^a{x'\'y)=xy (2). 

Whence,  by  comparing  equation  (1)  with  equation  (2). 
there  will  arise 

Where 
a-tp 

^        a+p         '^• 
And,  if  these  values  be  now  substituted  for  x-^-y  and  y  in 
equation  (2),  the  result,  when   simplified  and  reduced,  will 
give 

(a-^p)x^-'p(a-h\p)x~  -  ^ap^ 
From   which  last  equation   and  the  value  of  3/,  above 
found,  we  shall  have 

and 

And,  if  the  sum  of  these  two  sides  be  taken  from  p,  the 
kesult  will  give 


KB—p-{x+y) 


Which   expressions   are,   therefore,   respectively   equai 
to  the  values  of  the  three  sides  of  the  triangle. 


PROBLEM  XIV. 


Given  the  perpendicular,  base,  and  sum  of  the  sides  ot 
an  obtuse  angled  plane   triangle  abc,  to  determine  the  two     J 
sides  of  the  triangle. 


ALGEBRA  TO  GEOMETRY.  301 

A 


B  CD 

Let  the  perpendicular  ad=/j,  the  base  bc  =  6,  the  sum 
«f  AB  and  Ac=^.s,  and  their  difference  =--x. 

Then,  since  half  the  difference  of  any  two  quantities 
added  to  half  their  sum,  gives  the  greater,  and,  when  sub- 
tracted, the  less,  we  shall  have 

AB=^(sH-x),  and  ac=^(s— a:). 

But,  by  Euc.   1,  47,  cd^— ac^ — ad^,  or  00=^^  }  i(«  — 

•«)'~P'  i  ;  and,  by  B.  11,   12,  ab2=bc2-{-ac2+2bcXcd  j 
whence 

sx~b'=2b^^i{s-^xy-p  ^. 

And  if  each  of  the  sides  of  this  last  equation  be  squared, 
there  will  arise,  by  transposition  and  simplifying  the  re 
suit, 

{s^-~b')x^=b\s'^b^)—4by,  or 

Whence,  by  addition  and  subtraction,  we  shall  have 

''=2"  2^^'- 7-^^' 

Which  are  the  sides  of  the  triangle,  as  was  required. 

PROBLEM  XV. 

It  is  required  to  draw  a  right  line  bpe  from  one  of  the 
angles  b  of  ^  given  square  bd,  so  that  the  part  fe,  inter- 
cepted by  DE  and  dc,  shall  be  of  a  given  length. 

D  d 


S0£ 


APPLlCAT[ON  OF 
A-  D        K 


Bisect  FE  in  n,  and  put  ab  or  Bc=a,  fg  or  ge=6,  ana 
BG=a:  ;  then  will  BE=x-^b  and  bk=x— -6. 

But  since,  by  right  angled  triangles,  ae^=be-— ab*,  we 
shall  have 

And,  because  the  triangles  bcf,  eab,  are  similar, 
EF  ;  Bc  : :  BE  :  AK,  or 

Whence,  by  squaring  each   side  ol  this    equation,  and 
arranging  the  terms  in  order,  there  will  arise 
x''-2{a'-j-b'}^''^b\2n''-fj"). 
Which  equation,  being  resolved  after  the  manner   of  a 
quadratic,  will  give 

^^^\a'+h'  taV(a-\-^b')]. 
And.  consequently,  by  adding  b  to,  or  subtracting  it  from 
this  last  expression,  we  shall  have 

BL^-y\a'--{-b^±a^{a--{'Ab-')]-\-b,  or 
BF=^{a^-\-b'±a^{a^+4b'~)]  ~b. 
Which    values,   by  determining   the   point  f,  or  f,  will 
satisfy  the  problem. 

Where  it  may  be  observed,  that  the  point  g  lies  in  the 
circumference  of  a  circle,  described  from  the  centre  r. 
with  the  radius  fg,  or  half  the  given  line. 

PROBLEM  XVI. 

The  perimeter  of  a  right  angled  triangle  abc,  and  the 
radius  of  its  inscribed  circle  being  given,  to  determine  the 
triangle. 


ALGEBRA  TO  GEOMETRY. 


303 


Let  the  perimeter  of  the  triangle  =p.  the  radius  od,  or 
OE,  of  the  inscribed  circle  ==.  At.— x,  and  bu  =  j/. 

Then,  since  in  the  ri^ht  an^l^d  trianii Us  a'  o,  afo,  of  is 
equal  to  of,  and  oa  is  co-rnnon,  af  will  also  be  equal  ae, 
or  X. 

And,  in  like  manner,  it  mav  be  shown,  that  bk  is  equal 
to  Bn,  or  y. 

But,  by  the  question,  and  Euc.  i,  47,  we  have 
(x-l-r)-f-(v-f '•)^-'a'^-l/)  ^7),  and 
(.7-hr)^+-(/y^r)^'=(j-f2/)% 

Or,  by  adding  the  terms  of  the  first,  and  squaring  those 
of  the  second, 

r(.r4-?/)     xy — r^. 
Hence,  since,  in  the  first  ofihese  equations,  i/=^cip  —  r) 
—  x,  if  this  value  be  sub>tuuted  t'or   y  in  the  second,  there 
will  arise 

Which  equation,  being  resolved  in  the  usual  manner, 
gives 

and 
y^\{¥>—'')^V)  \{\P  -  rf—<lp-r)\. 
And,    consequently,  if  r  be  added  to    each  of  these  last 
expressions,  we  shall  have 

and 

for  the  values  ot  the  perpendicular  and  base  of  the  trian- 
gle, as  was  required. 


<)04 


APPLICATION  OF 


PROBLEM   XVll. 

From  one  of  the  extremities  a,  of  the  diameter  of  a  giv 
en  semicircle   adb    to    draw  a    right   hne  ae,   so   that  the 
part   PK,  intercepted  by  the  circuru  fere  nee  and    a  perpen- 
dicular drawn  from  the  other  extremity,  shall  be  of  a  given 
length. 

[C 


A  B 

Let  the  diameter  ab  — r/,  DE=rt,  and  AE=r;  and  join  bd. 
Then,  because  the  angle  adb  is  a  right  angle,  (Euc.  in, 
31.)  the  tnanules  abe,  abi>  are  similar. 

And,  consequently,   by  couiparing  their  like   sides,  we 
shall  have 

AE  :  AB  : :  AB  :  AD,  or 
rr  :  f/  :  :   d  :   or—o. 
Whence  multiplying  the  means   and   extremes   of  these 
proportionals,  there  vvill  arise 

s~ — aT=cl^, 
Which  equation,  being  resolved  after  the   usual  manner, 
gives 

PROBLEM  XVin. 

To  describe  a  circle  through  two  given  points  a,  b,  tha'c 
ghall  touch  a  right  line  cd  given  in  position. 

A. 


C    F 


ALGEBRA  TO  GEOMETRY.  305 

Join  AB  ;  and  through  o,  the  assumed  centre  of  the  re- 
quired circle,  draw  fe-  perpendicular  to  ab  ;  which  will 
bisect  it  in  k  (Euc.  in,  3). 

Also,  join  OB  ;  and  draw  eh,  og,  perpendicular  to  cd  ; 
the  latter  of  which  will  fall  on  the  point  of  contact  g  (Euc. 
ill,  18). 

Hence,  since  a,  e,  b,  h,  f,  are  given  points,  put  EB=a, 
EP  =  6,  EH--— c,  and  eo=.t  ;  which  will  give  of=/; — x. 

Then,  because  the  triangle  oeb  is  right  angled  at  e,  we 
shall  have 

0B^=Eo^-4-KB^,  or 
0B=-v/(j^4-a^). 

But,  by  similar  trianjifles,  fe  :  eh:*,  fo  :  of  or  gb  :  or 
&  :  c  : :  6  —  a; :  OB  ;  whence,  also. 

And,  consequently,  if  these  two  values  of  ob  be  put 
equal  to  each  other,  there  will  arise. 

Or,  by  squaring  each  side  of  this  equation,  and  simpli- 
fying the  result, 

(6^  -  c")a;2+26clr=62(c^-  a^). 

Which  last  equation,  when  resolved  in  the  usual  man- 
ner, gives 

for  the  distance  of  the  centre  o  from  the  chord  ab  ;  where 
h  must,  evidently,  be  greater  than  c,  and  c  greater  than  a. 

PROBLEM  XIX. 

The  three  lines  ao,  bo,  co,  drawn  from  the  angular 
points  of  a  plane  triangle  abc,  to  the  centre  of  its  inscri1)ed 
circle,  being  given,  to  find  the  radius  of  the  circle,  and  the 
sides  of  the  triangle. 

Dd2 


306 


B  EC 

Let  o  be  the  centre  of  the  circle,  and,  on  ao  produced 
let  fall  the  pefpc  ndiculars  cd  ;  and  drav  oe,  of,  og,  to  the 
points  of  contact  E,  f,  g. 

Then,  because  the  three  angles  of  the  triangle  aec  are- 
together,  equal  to  two  right  angles,  (Euc.  i,  32.)  the  sum 
of  their  halves  oac+oca-tobe  will  be  equal  to  one  right 
angle. 

But  the  sum  of  the  two  former  of  these,  OAc-f-ocA,  i.^ 
equal  to  the  external  angle  doc  ;  whence  the  sum  of  do( 
-J-OBfc:,  as  also  of  d«)c4-ocd,  is  equal  to  a  right  angle  :  and. 
consequently,  obk=ocd. 

Let,  therefore,  AO=a,  bo  =  6,  co  =  c,  and  the  radius  oe. 
OF,  or  oG=a:. 

Then,  since  the   triangles    boe,  con,    are  similar,  bo 
OE  :  :  CO  :  on,  ox  b  :  .r  :  :  c  :  on  ;   which  gives 

COC  C"X~  c 

cD=-j-,  and  CD=  v/  (c" — rV)  oj"  t\/C^"  —  '''''^)' 
b  b~  o 

Also,  because  the  triangle  aoc  is  obtuse  angled  at  o,  \vf 

■shall  have  (Euc.  n,  12.) 

ac^=ac^4-co^+2aoXod  ;  or 

AC=^(a2H-c^H —)  or^( ^ ). 

But  the  triangles  ago,  aof,  being  likewise  similar, 
AC  :  CD  ;  :  AO  :   of,  or 

'   Whence,  multiplying  the  means  and  extremes,  and  squar- 
mg  the  result,  there  will  arise 

hx'l  h  (aH  c";  -f  2acx  ]  =a^c\b'-^  x~). 
Or,   by  collecting  the  terms  together,  and  dividing  b^ 
the  coefficient  of  the  highest  power  of  xt 


ALGEBRA  TO  GEOMETRY.  307 

From  which  last  «quaiion  x  may  be  determined,  and 
ihence  the  side  of  the  triajigle.* 

PROBLEM  XX. 

Given  the  three  -ides  ab,  bc,  cd,  of  a  trapezium  abcd, 
inscribed  in  a  semicircle,  to  find  the  diameter,  or  remain- 
injT  side  ad. 


Let  AB=a,  BC=ft,  CD=c,  and  ad=^x  ;  then,  by  Euc.  vi. 
D,  acXbd=adXb    -^ ABXcD=bx-{-ac. 

But   ABD,  acd,  being  right  angles,  (Euc.   iii,  31,)   we 
shall  have 

Ac~  ^(ad-~dc^),  or  \/{x^  ~c^),  and 
bd==-v/(ad"  -  ab^),  or  y^(i^  — a^). 
Whence,  by  substituting  these  two  values  in  the  former 
expression,  th<re  will  arise 

^(x'-i-)  X  ./(.r'-  a2)=6x-f-ac. 
Or,  by  squaring  each  side,  and  reducing  the  result. 

From  which  last  equation  the  value  of  a  may  be  found, 
as  in  the  last  problem.! 


*  This,  and  the  following;  problem,  cannot  be  constructed  geometricallj, 
or  by  means  onl}'  of  rig;ht  lines  and  a  circle,  being  what  the  ancients  usuall/ 
denominated  solid  problems,  from  the  circumstance  of  their  involving  aa 
equation  of  more  than  two  dimensions  ;  in  which  cases  they  generallj  era- 
ployed  the  conic  sections,  or  some  of  the  higher  orders  of  curves. 

f  Newton,  in  his  Universal  Arithmetic,  English  edition,  1728,  has  resolved 
this  problem  in  a  variety  of  different  ways,  in  ofder  to  shonr  that  some  me- 
thods of  proceeding,  in  cases  of  this  kind,  frequently  lead  tomore  elegant  so- 
lutions than  others ;  and  that  a  ready  knowledge  of  these  can  only  be  obtaitt " 
ed  by  practice. 


(308) 
MISCELLANEOUS  PROBLEMS. 


PROBLEM  1. 


To  find  the  side  of  a  square,  inscribed  in  a  given  semi- 
circle, whose  diameter  is  d. 

Ans.  -=dy/h' 


PROBLEM  11. 

Having  given  the  hypothenuse  (1 3)  of  a  rij^ht  angled 
triangle,  and  the  difTerence  between  the  other  two  sides 
(7),  to  find  these  sides*.  Ans.  5  and  12. 

PROBLEM  III. 

To  iind  the  side  of  an  equilateral  triangle,  inscribed  in 
a  circle  whose  diameter  is  d  ;  and  that  of  another  circum- 
scribed about  the  same  circle. 

Ans.  i(/v/3,  and  (/-/3. 

PROBLEM  IV. 

To  find  the  side  of  a  regular  pentagon,  inscribed  in  a 
circle,  whose  diameter  is  d.  Ans.  ^f/y^(10--2^5). 

PROBLEM  V. 

To  find  the  sides  of  a  rectangle,  the  perimeter  of  which 
.shall  be  equal  to  that  of  a  square,  whose  side  is  a,  and  its 
area  half  that  of  a  square.    Ans.  a-\-\ay/2  and  a — \a»yi, 

PROBLEM  VI. 

Having  given  the  side  (10)  of  an  equilateral  triangle, 
to  find  the  radii  of  its  inscribed  and  circumscribing  circles. 

Ans.  2.8^68  and  5.7736. 


*  Such  cf  these  ques'ions  as  are  proposed  in  numbers,  ehould  first  be  re- 
solved generally,  by  mean? of  the  usual  symbols,  and  then  reduced  to  theari- 
pwers  above  given,  by  substituting  the  nunieral  values  of  the  letters  in  the 
results  thus  obtaiued. 


MISCELLANEOUS  PROBLEMS.  309 


PROBLEM  VII. 

Having  given  the  perimeter  (12)  of  a  rhombus,  and  the 
sum  (8)  of  its  two  diagonals,  to  find  the  diagonals. 

Ans.  4-i--v/2  and  4-v^2. 

PROBLEM  VIII. 

Required  the  area  of  a  right  angled  triangle,  whose  hy- 
pothenuse  is  .r^%  and  the  base  and  perpendicular  a;''^  and 
xr.  Ans.  1.0^9085, 

PROBl>EM  IX. 

Having  given  two  coniiguous  sides  (a,  b)  of  a  paral- 
lelogram, and  one  of  its  diagonals  (<i),  to  find  the  other 
diagonal.  Ans.  ^{ta'+'ib^-d^)* 

PROBLKM  X. 

Having  given  the  perpendicular  (300)  of  a  plane  trian- 
gle, the  sain  of  the  two  sides  (^iloO),  and  the  ditFerence 
of  the  segments  of  tne  base  (495),  to  find  the  base  and  the 
sides.  Ans.  945,  ^^75,  and  780. 

PKOBLUVl  XI. 

The  lengths  of  three  lines  drawn  from  the  three  angles 
of  a  plane  triangle  to  the  .niddle  of  the  opposite  sides,  be- 
ing 18,  "14,  and  jO,  respectively  :  it  is  required  to  find  the 
sides.  Ans.  "^U,  28.84  i,  and  34.176. 

PROBi  EM  Xll. 

In  a  plane  triangle,  there  is  given  the  base  (50),  the 
area  (7yb),  and  the  difierence  of  the  sides  (lO),  to  find 
the  sides  and  the  perpendicular.     Ans.  36,  46,  and  33.26L 

PROBLEM  XIII. 

Given  the  base  (!94)  of  a  plane  triangle,  the  line  that 
bisects  the  vertical  angle  (66),  and  the  diameter  200)  of 
the  circumscribing  circle,  to  find  the  other  two  sides. 

Ans.  81.36587  and  167.43865, 


310  IVIISCELLANEOUS  PROBLEMS. 

PHMBLf  M   XIV. 

The  lenorths  of  two  line-  th-»t  bisect  the  acute  angles  of 

aright  angled  plane  triai.yie.  l>ring  40  anri  5(  respectively^ 

it  is  required  to    determiMK  the  three  side>  n\'  the  triangle. 

Ans.  d5.bn7.37,  47.4U72b,  and  59.41143. 

PhOll.E.M   XV. 

Given  the  altitude  (4)  the  ba-^e  (8),  and  the  sunn  of  the 
sides  (12),  of  a  plane  triangle,  to  find  the  sides. 

4  4 
Anri.  6+ ^^5  and  6--v/5. 

5  5 

PROBLEM  XVI. 

Having  given  the  base  of  a  plane  triangle  (15),  its  area 
(45).  and  the  ratio  of  its  other  two  side-  as  2  to  3,  it  is  re- 
quired to  deterriiine  the  lengths  of  the.-se  >ides. 

Ans.  :.;yl5  and  11.6872. 

PROBLl-M  XVn. 

Given  the  perpendicular  (24)  the  like  bisecting  the 
base  (40),  and  the  line  bisecting  the  vertical  angle  (^5),  to 
determine  the  triangle. 

250 
Ans.  The  base  -  -v/7 
7  ^ 

From  which  the  other  two  sides  may  be  readily  found. 

PKOBLEM  XVlll. 

Given  the  hypothenuse  (10)  of  a  right  angled  triangle, 
and  the  difference  of  two  lines  drawn  fronj  its  exlremiiies 
to  the  centre  of  the  inscribed  circle  (2),  to  determine  the 
base  and  perpendicular. 

Ans   8.0ru04  and  5.87447. 

PROBLEM  XIX. 

Having  given  the  lengths  (a,  6.)  of  two  chords,  cutting 
each  other  at  right  angles,  in  a  circle,  and  the  distance  (c) 


MISCELLANEOUS  PROBLEMS.  3U 

•i*  their  point  of  intersection  from  tiie  centre,  to  determine 
the  diameter  of  a  circle. 

Ans.  ^{8{a'-\'b^)i-2c'l, 

PR0BLE3I  XX. 

Two  trees,  standing  on  a  horizontal  plane,  are  120  feet 
asunder  ;  the  height  of  the  highest  of  which  is  100  feet, 
and  that  of  the  shortest  bu  ;  whereah«»iits  in  the  plane 
must  a  person  place  himstlf,  so  that  his  distance  from  the 
top  of  each  tree,  and  the  distance  of  the  tops  themselves, 
shall  be  all  equal  to  each  other? 

Ans.  2Uv^2  i  feet  from  tfie  bottom  of  the  shortest, 
and  40^3  feet  from  the  bottom  of  the  other. 

PROBLEM  XXi. 

Having  given  the  sides  of  a  trapezium,  inscribed  in  a 
circle,  equal  to  6,  4,  5,  and  3,  respectively,  to  determine 
the  diameter  of  the  circle. 

Ans.  i-v/(130X153)  or  7.051595. 

PROBLEM  XXII. 

Supposing  the  town  a  to  be  30  miles  from  b,  b  25  miles 
from  c,  and  c  20  miles  from  a  ;  whereabouts  must  a  house 
be  erected  that  it  shall  be  at  an  equal  distance  from  each 
of  them  ?  Ans.  15. 1 18556  miles  from  each. 

PROBLEM  XXIll. 

Given  the  area  (100)  of  an  equilateral  triangle  abc, 
whose  base  bc  falls  on  the  diameter,  and  vertex  a  in  the 
middle  of  the  arc  of  a  semicircle  ;  required  the  diameter 
of  the  semicircle.  Ans.  20^3. 

PROBLEM  XXIV. 

In  a  plane  triangle,  having  given  the  perpendicular  (p), 
and  the  radii  (r,  r)  of  its  inscribed  and  circumscribing  cir- 
cles, to  determine  the  triangle. 

Ans.  The  base  ^rVlfcirSr-a 
p — 2r 


312  MISCELLANEOUS  PROBLEMS. 

PROBLEM  XXV. 

Having  given  the  base  of  a  plane  triangle  equal  to  isia, 
the  perpendicular  equal  to  a,  and  the  sum  of  the  cubes  of 
its  other  two  sides  equal  to  three  times  the  cube  of  the 
fease ;  to  determine  the  sides. 


Ans.  a{2'\-l^6)  and  ct{2'^\y/e) 


THE  END. 


6  3  7 


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